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The finite field restriction problem Sollazzo, Rhoda Jane 2011

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The Finite Field Restriction Problem by Rhoda Sollazzo B.Sc. Pure and Applied Mathematics, Concordia University, 2009 a thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in the faculty of graduate studies (Mathematics) The University Of British Columbia (Vancouver) August 2011 c© Rhoda Sollazzo, 2011 Abstract This work studies the extension problem for subsets of finite fields. This remains an important unsolved problem in harmonic analysis, in both the Euclidean and finite field setting. We survey the partial results obtained to date, common techniques, and open conjectures. In the case of a homogeneous variety H over a d-dimensional finite field, the L2 → L4 boundedness is proved whenever H contains no hyperplanes. This is accomplished by proving an incidence theorem for cones of this type, and applying a sufficient condition for L2 → L2m obtained by Mockenhaupt and Tao in their 2004 introductory paper. We moreover present counterexamples for particular cones when r < 4, establishing the 2 to 4 bound as optimal in r for general homogeneous varieties. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Notation and Definitions in the Finite Field . . . . . . . . . . . . 3 1.2 Characters and Gauss Sums . . . . . . . . . . . . . . . . . . . . . 5 1.3 Basic Inequalities and Identities . . . . . . . . . . . . . . . . . . . 8 1.3.1 Riesz-Thorin interpolation . . . . . . . . . . . . . . . . . . 12 1.3.2 Young’s inequality . . . . . . . . . . . . . . . . . . . . . . 12 2 Progress Thus Far . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1 Some Trivial Estimates . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Necessary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Tomas-Stein Estimates . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.1 Tomas-Stein type argument for even exponents . . . . . . 21 2.3.2 A Tomas-Stein type estimate for the cone . . . . . . . . . 24 2.4 Dyadic Pigeonholing and Incidence Arguments . . . . . . . . . . 27 2.4.1 The dyadic pigeonholing argument . . . . . . . . . . . . . 29 2.5 Summary of Existing Results . . . . . . . . . . . . . . . . . . . . 30 2.5.1 Paraboloids . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.5.2 Quadratic Surfaces . . . . . . . . . . . . . . . . . . . . . . 31 2.5.3 Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 iii 2.5.4 Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3 Restriction Estimates for Cones over Finite Fields . . . . . . . 33 3.1 Preliminary Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 Proof of Theorems 3.0.1 and 3.0.2 . . . . . . . . . . . . . . . . . 36 3.2.1 Proof of Theorem 3.0.1 . . . . . . . . . . . . . . . . . . . 36 3.2.2 Proof of Theorem 3.0.2 . . . . . . . . . . . . . . . . . . . 37 3.3 Possible Improvements . . . . . . . . . . . . . . . . . . . . . . . . 38 3.4 Upper Bounds on r for Specific Cones . . . . . . . . . . . . . . . 38 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 iv List of Tables Table 2.1 Summary of Existing Results for p→ r. . . . . . . . . . . . . 32 v Acknowledgments I would like to express my deep gratitude and appreciation to Dr. Izabella Laba and Dr. Mahta Khosravi for their supervision over the course of this degree. For their continual support, generous encouragement, and sound instruction, I count myself incredibly fortunate. I could not have hoped for more welcoming, exhortative and constructive supervisors. I thank NSERC for their financial support via the PGS Fellowship, which has afforded me time to focus on this research. This thesis could would not have been possible without the overall support of the UBC Mathematics Department. Particular thanks are owed to graduate advisor Greg Martin for his patient and sound advice, to graduate secretary Lee Yupitun for her cheerful assistance, as well as the Math Department front office staff, whose friendly banter and helpful dispositions have brightened my day on many occasions. I thank my parents for their patience with my prolonged absence, unreliability on the telephone, and esoteric descriptions of my work. Thank you for the support, the rides to the airport, the concern, and the phone calls. Whatever success I have had in graduate school is propped up by the won- derful people I have met at Green College, and the beautiful ideas and friendship I have encountered there. I owe a great debt to friends who have kept me sane during studying sessions, held me upright through tumultuous times, worked, played, laughed and cried with me over the last two years — Vahid Bazargan, Fabiola Carletti, Emily Davidson, Ali Kaufman, Jessica Lubrick, Andrew Mac- Donald, Michael Muthukrishna, Amanda Perry, Sigal Samuel, Claire Stilwell, Alyssa Stryker, Wanying Zhao. Thank you, and so many others, for making my life here the priceless adventure it has been. To those many people who are owed my gratitude but omitted from this list, vi please accept my apologies, and a deferral of my heartfelt thanks. I consider myself unfairly blessed to have you all in my life. I have every intention to make that clear — in our time together if not through this note. U.I.O.G.D. vii Soli Deo gloria. viii Chapter 1 Introduction In the 1960s, Elias Stein posed a question concerning Fourier transforms of func- tions which have been restricted to a subset of Euclidean space, which is still unsolved. The classical version of this problem can be roughly thought of as an analysis of the boundedness of the Fourier transform operator f̂ on Rd, for different classes of functions f . If f is an L1 function, we know from the Riemann- Lebesgue lemma that its Fourier transform f̂ is continuous on Rd, is bounded, and vanishes at infinity [3]. If we restrict f to some subset E ⊂ Rd (and denote this restriction by f ∣∣ E ), in this case we know f̂ ∣∣ E to be a continuous bounded function on E. If we consider the class of L2 functions, on the other hand, f̂ need not be continuous. From Plancherel’s identity, we know that f̂ will be an L2 function, but it is not necessarily defined everywhere. This means we cannot generally restrict such functions to a measure zero set. We turn our attention to the intermediary cases, where f ∈ Lp for some 1 < p < 2. It is unclear how much can be said about the behaviour of the Fourier transform in this case - the Fourier transform here need not be continuous or bounded, but we may still have some form of good decay. The Hausdorff-Young inequality (1.6) tells us that when f ∈ Lp with 1 < p < 2, f̂ lies in the dual space Lp ′ , where 1p + 1 p′ = 1 (so 2 < p < ∞). We can certainly restrict f̂ to any set E of positive measure. On the other hand, if E is a set of zero measure, it is not clear what we can expect. This leads to the central motivating question, initially formulated by Elias Stein: For which values of p ∈ [1, 2] and which sets E ⊂ Rd may the Fourier 1 Transform of a generic Lp(Rd) function be restricted to E? We can reformulate this question by seeking a restriction estimate of the following form. Let S be a compact subset of some E ⊂ Rd. We seek values of 1 < p < 2 and 1 ≤ r ≤ ∞ such that for every Schwartz function f we have∥∥∥f̂ ∣∣S∥∥∥Lr(S,dσ) . ‖f‖Lp(Rd) . Here and throughout, A . B denotes that there exists some constant C such that A ≤ CB. The constant here may depend only on p, r and S. We denote the surface measure on S by dσ . In the Euclidean setting this is generally the usual Lebesgue measure on the set. If such an estimate holds, we may use the density of the class of Schwartz functions in L2 to obtain a restriction operator from Lp(Rd, dx) to Lr(S, dσ) for general functions in Lp. By duality and Parseval’s identity (see Remark 1), we may equivalently seek values of p and r such that for all smooth functions f on S we have ‖(fdσ)̌‖Lp′ (Rd,dx) . ‖f‖Lr′ (S,dσ) . This formulation of the problem is often referred to as the extension problem, as we take a function defined on S and attempt to extend the inverse Fourier transform to Rd. This has become the more common expression of the problem in recent years, owing to numerous applications to partial differential equations. Stein showed that such estimates do exist for non-trivial values of p and r, when the subset E ∈ Rd is a surface with sufficient curvature. The classical case of the restriction problem considers the sphere in Rd. Partial results are known for a variety of sets, most notably spheres, cones and paraboloids. It is conjectured that an estimate exists for spheres and paraboloids if and only if p < 2d d+ 1 , and (d+ 1) p′ < d− 1 r . This is known when d = 2, but remains open for higher dimensions. The best known result to date is due to Tao [16]. The bilinear version of this result has been obtained by Wolff [20]. The necessary and sufficient conditions for cones 2 are conjectured to be p < 2d− 1 d , d p′ < d− 2 r . In 2003, Gerd Mockenhaupt and Terence Tao initiated a second phase of work on this question, which relocates the base field from d-dimensional Euclidean space to a vector space over a finite field in d-dimensions. The finite field set- ting provides us with a model of the Euclidean case that is more accessible in some respects. The finite field version of the problem is additionally connected to existing problems in algebraic geometry, combinatorics and additive number theory, so is of interest in its own right. The restriction problem is suspected to be more tractable in the finite setting, as we have access to the discrete meth- ods offered by these fields. On the other hand, several techniques native to the Euclidean case are lost, for example Taylor expansions and dyadic scalings. In 2008, a similar adaptation of the Kakeya problem to finite fields was successfully solved by Zeev Dvir, using a polynomial method [2]. The Euclidean Kakeya conjecture is known to imply the Euclidean restriction conjecture. This does not hold it the finite case, and the finite field restriction problem has only been solved for particular low-dimension cases, as with its Euclidean counterpart. It is thought that further developments in the finite field setting will assist in our understanding of the Euclidean case. In what follows we will express the finite field version of the problem formally, survey known methods and results, and present some new results for the cone in finite fields. We begin by introducing some basic notation and definitions for our problem. 1.1 Notation and Definitions in the Finite Field Let F be a finite field with q elements, and let d be the dimension of space. We write F∗ for the multiplicative group F \ {0}, and F∗ for the dual space of F. We denote the cardinality of a set E in Fd by |E|. For any abelian finite group G, we call a function χ : G → {z ∈ C : |z| = 1} a character if it is homomorphic; i.e. χ(g1 · g2) = χ(g1)χ(g2), for all gi ∈ G. The set of all characters on G is itself an abelian group of size |G|, with respect to multiplication. We denote the group of characters over G by G∧. Since we are working over a field, we can consider characters over the additive group F+ and over the multiplicative group F∗. We generally denote additive and multiplicative 3 characters by χ and ψ, respectively. Relevant results on characters are detailed in the following section. Let f(x) be a complex-valued function on Fd. Over finite fields integrals are replaced by finite sums, so that when dx is a counting measure we have∫ E f(x)dx = ∑ x∈E f(x). We consider complex valued functions f(x) defined on Fd. In this context, we define the Fourier transform of f as f̂(ξ) := ∫ Fd f(x)χ(−x · ξ)dx = ∑ x∈Fd f(x)χ(−x · ξ). Here dx is the counting measure on Fd, x · ξ denotes the inner product x · ξ = d∑ i=1 xiξi, and χ is some non-trivial additive character (we may think of this as the equivalent of the function e2piix in the Euclidean case). Note that f̂ is defined on the dual space of Fd. We shall denote the dual space by Fd∗, and equip it with the normalized counting measure dξ . Integration over the dual space becomes∫ Fd∗ g(ξ)dξ := 1 qd ∑ ξ∈Fd∗ g(ξ). As in the continuous case, we have an inversion formula for the Fourier transform: f (̌x) := ∫ Fd∗ f(ξ)χ(x · ξ)dξ = 1 qd ∑ ξ∈Fd∗ f(ξ)χ(x · ξ). For 1 ≤ p <∞ the Lp norm of f over Fd is given by ‖f‖Lp(Fd,dx) = ( ∫ Fd |f(x)|pdx) 1p = ( ∑ x∈Fd |f(x)|p) 1p . When p =∞, we have ‖f‖L∞(Fd,dx) = sup x∈Fd |f |. Note that Lp spaces are nested over fields of finite measure, as we will prove in the following section (Theorem 1.3.2). Let S be a surface in Fd. We endow S with a normalized surface measure dσ, 4 given by dσ(x) = qd |S|1Sdx, where 1S is the characteristic function of S. This normalizes our integration such that the surface has total mass equal to one. For a function f restricted to S, this yields ‖f‖Lp(S,dσ) = ( ∫ S |f(x)|pdσ(x)) 1p = ( 1|S| ∑ x∈S |f(x)|p) 1p . and the corresponding inverse transform (fdσ)̌ (ξ) = ∫ S χ(ξ · x)f(x)dσ(x) = 1|S| ∑ x∈S χ(ξ · x)f(x). We express the restriction problem in this context as follows: given a subset E of Fd, for which values of 1 ≤ p, r ≤ ∞ do we have an estimate of the form ‖(fdσ)̌‖Lr(Fd,dξ) . ‖f‖Lp(S,dσ) , (1.1) independent of p and r, for every complex-valued function f on Fd and every subset S of E. By duality (Remark 1), we can equivalently express this as the estimate ∥∥∥f̂∥∥∥ Lp′ (S,dσ) . ‖f‖Lr′ (Fd,dξ) (1.2) where p′ and r′ are again used to denote dual (conjugate) exponents. Let R(p → r) denote the smallest constant for which (1.1) holds. Note by duality that this is also the best constant under which (1.2) holds. The finite field restriction problem then seeks the values of p and r for which we have R(p→ r) . 1. 1.2 Characters and Gauss Sums Over finite fields, the exponential function in the Fourier transform is replaced by a non-trivial character - we will often rely on results on character sums in order to obtain bounds on the norm of f̂ . We collect some basic definitions and relevant results on characters here. For proofs of these results, and further discussion of exponential sums, see [12]. 5 For any character over a finite abelian group (G, ·), we have χ(x)χ(y) = χ(x · y), for all x, y ∈ G. Let 1G denote the identity in (G, ·).When χ(x) := 1G for all x ∈ G, we call χ the trivial character (often denoted χ0). For every character we have χ(1G) = 1, since χ(1G)χ(x) = χ(1G · x) = χ(x). If G is a cyclic abelian group, with g as a generator, we have characters of the form χj(g k) := e 2piijk q for every integer 0 ≤ j < q. In fact, in this case G∧ consists exactly of the characters in this form. We call the smallest positive integer k such that (χ(x))k = 1 for all x the order of χ. Since G is finite, note that for all χ ∈ G∧ and x ∈ G we have (χ(x))|G| = χ(x|G|) = χ(1G) = 1, so every character is of finite order. When k = 2, χ is said to be a quadratic character. Let ℘ be the characteristic of the finite field F. Then the prime field F℘ of size ℘ (which is unique up to isomorphism) is contained in F. Let Tr be the trace function from F to F℘. Since the trace function is an additive homomorphism, the function χ1(x) := e 2piiTr(x) ℘ is a character of F+. We call this the canonical additive character. Lemma 1.2.1 (Basic Character Sums). Let G be a finite abelian group, and let G∧ be the group of all characters over G. Then we have ∑ x∈G χ(x) = 0 if χ is non-trivialq if χ = χ0 and ∑ χ∈G∧ χ(x) = 0 if x 6= 1Gq if x = 1G . Proof. First, note that when χ = χ0, we have ∑ x∈G χ0(x) = ∑ x∈G 1, and when x = 1G, ∑ χ∈G∧ χ(1G) = ∑ χ∈G∧ 1 = |G|. If χ 6= χ0, then there exists some y ∈ G such that χ(y) 6= 1. But χ(y) ∑ x∈G χ(x) = ∑ x∈G χ(y)χ(x) = ∑ x∈G χ(yx) = ∑ yx∈G χ(yx) = ∑ x∈G χ(x), so χ(y) 6= 1 implies ∑ x∈G χ(x) = 0. 6 Now, for a fixed non-identity x ∈ G, define the function x̂ such that x̂(χ) = χ(x), for all χ ∈ G∧. This is a non-trivial character on G∧, since x̂(χ)x̂(χ′) = χ(x)χ′(x) = χχ′(x) = x̂(χχ′), and there exists some χ for which χ(x) 6= 1. Then applying previous result, we have ∑ χ∈G∧ χ(x) = ∑ χ∈G∧ x̂(χ) = 0. An application of these sums yields the following orthogonality relations: Theorem 1.2.2 (Orthogonality Relations for Characters). Let χa, χb be multi- plicative characters over finite abelian group G. Then for every c, d ∈ G, we have the following relations: 1. ∑ c∈G χa(c)χb(c) = 0 if a 6= b|G| if a = b 2. If a 6= 0 (i.e. χa is non-trivial), then ∑ c∈F ∗ χa(c) = 0 3. ∑ χ∈G∧ χ(c)χ(d) = 0 if c 6= d|G| if c = d Since we are working over a field, it will often be useful to consider sums over the product of multiplicative and additive characters. For an additive character χ and multiplicative character ψ, a sum of the form ∑ x∈F∗ χ(x)ψ(x) is called a Gaussian (or Gauss) sum. The above relations allow us to evaluate this sum as follows. Theorem 1.2.3. Let G(χ, ψ) = ∑ x∈F∗ χ(x)ψ(x) be a Gauss sum. Then G evaluates to G(χ, ψ) =  |F| − 1 if χ = χ0, ψ = ψ0 −1 if χ = χ0, ψ 6= ψ0 0 if χ 6= 0, ψ = ψ0 We also note that |G(χ, ψ)| ≤ |F| − 1 for every choice of characters χ and ψ. If χ 6= χ0, ψ 6= ψ0, we have |G(χ, ψ)| = |F| 12 . When η is a quadratic multiplicative character, for every y ∈ F we have the Gauss sum Gy(χ, η) = ∑ x∈Fd χ(xy)η(x) 7 with size |Gy(χ, η)| = q 1 2 if y 6= 0 0 if y = 0 When y = 1, we obtain the following character sum: Theorem 1.2.4. Let η be a quadratic multiplicative character, and let χ be the canonical additive character on F. Then for every a ∈ F∗ we have character sum∑ s∈F χ(as2) = η(a)G1(η, χ). Theorem 1.2.5. Let η and χ be quadratic and canonical characters respectively, on F d. Let t ∈ F , and α, β ∈ Fd. Then ∑ α∈Fk χ(tα · α+ β · α) = χ(‖β‖2−4t )η k(t)(G(η, χ))k (1.3) Theorem 1.2.6. Let ψ be a non-trivial additive character over F, then | ∑ ξ,η∈F\0 ψ(xξ2)ψ(−xη2)| = |F|. (1.4) 1.3 Basic Inequalities and Identities The finite field setting retains many of the inequalities and identities we make use of in the continuous case. The proofs in this setting often become simple ma- nipulations of sums and applications of the orthogonality relation for characters. We collect some of these results here. Lemma 1.3.1. Let f, g be complex-valued functions on Fd. Then we have (i) Parseval’s identity: ∫ Fd f(x)g(x)dx = ∫ Fd∗ f̂(ξ)ĝ(ξ)dξ (ii) Plancherel’s identity:‖f‖L2 = ∥∥∥f̂∥∥∥ L2 Proof. (i) Writing out the right hand side in full and rearranging the integrals 8 yields ∫ Fd∗ f̂(ξ)ĝ(ξ)dξ = ∫ Fd∗ ∫ Fd f(x)χ(−x · ξ)dx ∫ Fd g(y), χ(y · ξ)dydξ = ∫ Fd ∫ Fd f(x)g(y) ∫ Fd∗ χ(−x · ξ)χ(y · ξ)dξdxdy = ∫ Fd ∫ Fd f(x)g(y)δ0(x− y)dydx (1.5) = ∫ Fd ∫ Fd f(x)g(x)dx where (1.5) is from Lemma 1.2.1. (ii) Set g = f in (i). We retain Hölder’s inequality, which states for all 1 ≤ p, p′ ≤ ∞ such that 1 p + 1 p′ = 1 we have ‖fg‖L1 ≤ ‖f‖Lp ‖g‖Lp′ or ( ∑ x∈Fd |f(x)g(x)|) ≤ ( ∑ x∈Fd |f(x)|p) 1p )( ∑ x∈Fd |g(x)|p′) 1p′ ) for all functions f and g on which this is well-defined. For a proof of this on generic measure spaces, see [14]. From this one can prove Minkowski’s inequality, which establishes the triangle inequality for Lp norms. For any 1 ≤ p ≤ ∞ we have ‖f + g‖p ≤ ‖f‖p + ‖f‖p We also have the Hausdorff-Young inequality∥∥∥f̂∥∥∥ Lp′ ≤ ‖f‖Lp , where f ∈ Lp. (1.6) We define the convolution of two complex-valued functions f, g on Fd as follows: (f ∗ g)(x) := ∫ Fd f(y)g(x− y)dy = ∑ y∈Fd f(y)g(x− y). Note that convolution is symmetric with respect to x and y. Moreover, we 9 calculate f̂ ∗ g(ξ) = ∫ Fd (∫ Fd f(y)g(x− y)dy ) χ(−x · ξ)dx = ∫ Fd f(y) ∫ Fd g(x− y)χ((−x+ y − y) · ξ)dxdy = ∫ Fd f(y)χ(−y · ξ) (∫ Fd g(x− y)χ(−(x− y) · ξ)dx ) dy = ∫ Fd f(y)χ(−y · ξ)ĝ(ξ)dy = f̂(ξ)ĝ(ξ) so we have (̂f ∗ g)(ξ) = f̂(ξ)ĝ(ξ) (1.7) and f̂g(ξ) = (f̂ ∗ ĝ)(ξ) It is also straightforward to calculate the inversion formula. We define the inverse Fourier transform of g : Fd∗ → C so that g (̌x) = ∫ Fd∗ g(ξ)χ(x · ξ)dξ, then we have (f̂ )̌ (x) = f(x), for every x ∈ Fd. Over finite fields we have a monotonic nesting of Lp norms. This is an im- portant distinction between the Euclidean and finite settings which we will make use of to develop some simple restriction results in Section 2.1. Theorem 1.3.2 (Nesting of Lp-spaces). Suppose 1 ≤ p1 ≤ p2 ≤ ∞. Let µx and µσ be a simple counting measure and a normalized counting measure respectively, on the finite space Fd. Then (1) ‖f‖Lp2 (Fd,µx) ≤ ‖f‖Lp1 (Fd,µx) (2) ‖f‖Lp1 (Fd,µσ) ≤ ‖f‖Lp2 (Fd,µσ) so we have Lp1(Fd, µx) ⊂ Lp2(Fd, µx) and Lp2(Fd, µσ) ⊂ Lp1(Fd, µσ). Proof. (1) Let f be such that ‖f‖Lp1 (Fd,µx) = 1. It suffices to show that 10 ‖f‖Lp2 (Fd,µx) ≤ 1 in this case. We have ‖f‖L∞ ≤ 1, and ‖f‖Lp2 = (∫ Fd |f |p2dµx ) 1 p2 = ∑ x∈Fd |f(x)|p2  1p2 = (∑ Fd |f |p1 |f |p2−p1dµx ) 1 p2 ≤ ‖fp2−p1‖L∞(Fd,µx) ∑ x∈Fd |f |p1  1p2 = ‖f‖ p2−p1 p2 L∞ ‖f‖ p1 p2 Lp1 ≤ ‖f‖ p1 p2 Lp1 = 1 (2) Since µσ is a normalized measure, we have ‖1‖Lp = 1, for all 1 ≤ p ≤ ∞. Applying Hölder’s inequality to the Lp1 norm, we see that ‖f‖Lp1 = (∫ Fd |f(x)|p1dµσ ) 1 p1 = (∫ Fd 1 · |f |p1dµσ ) 1 p1 ≤ ((∫ Fd 1 p2 p2−p1 dµσ ) p2−p1 p2 (∫ Fd |f |p1 p2 p1 µσ ) p1 p2 ) 1 p1 = |1‖ p2−p1 p2 Lp1 (∫ Fd |f |p2dµσ ) 1 p2 = ‖f‖Lp2 , since we have a normalized measure. Remark 1. If we define the operator T from functions on Fd to those on E such that Tf := f̂ ∣∣ E , then the adjoint operator T ∗ is given by the extension map T ∗ : g 7→ (gdσ)̌ . This follows easily from Parseval’s identity — we note that < f, T ∗g >= ∫ Fd f ( gdσ ) ˇ = ∫ Fd f̂ ( gdσ ) = ∫ Fd f̂ ∣∣ E gdσ =< Tf, g >. This 11 establishes the equivalence of our dual statements (1.1) and (1.2), and will also allow us to develop the Tomas-Stein argument for finite fields in Section 2.3. 1.3.1 Riesz-Thorin interpolation Riesz-Thorin interpolation provides one of the principal tools for developing ranges of exponents p and r. When two estimates are known for a given subset E, we may interpolate between the two sets of exponents in order to produce a range of p and r for which (1.2) will hold. Riesz-Thorin Interpolation is a result on generic linear operators, and we present it in its general form before turning to our particular case. A proof can be found in [8]. Theorem 1.3.3 (Riesz-Thorin Interpolation). Let 1 ≤ p1, p2, r1, r2 ≤ ∞. For all 0 < θ < 1 we define p, r such that 1 p = 1− θ p1 + θ p2 and 1 r = 1− θ r1 + θ r2 Let T : (Lp1 + Lp2)→ (Lr1 + Lr2) be a linear operator such that ‖Tf‖Lr1 ≤ C1 ‖f‖Lp1 for every f ∈ Lp1 and ‖Tf‖Lr2 ≤ C2 ‖f‖Lp2 for every f ∈ Lp2 then ‖Tf‖Lr ≤ C1−θ1 Cθ2 ‖f‖Lp for every f ∈ Lp. 1.3.2 Young’s inequality We can bound the convolution of f and g by norms of the original functions according to the following inequality. We will make use of this fact to develop Tomas-Stein type results in Section 2.3. Theorem 1.3.4 (Young’s inequality). Let f ∈ Lp(Fd), g ∈ Lr(Fd) be complex- valued functions. Let s be such that 1 p + 1 r = 1 s + 1. 12 then we have ‖f ∗ g‖Ls(Fd,dx) ≤ ‖f‖Lp(Fd,dx) ‖g‖Lr(Fd,dx) Proof. Let f ∈ Lp(Fd). The inequality is easily obtained via Reisz-Thorin in- terpolation of the operator given by convolution with f . We define T such that Tg := f ∗ g, and interpolate the following two bounds: (i) Let g ∈ Lp′(Fd). Then f ∗ g ∈ L∞ and ‖Tg‖L∞(Fd,dx) ≤ C1 ‖g‖Lp′ (Fd,dx) (ii) Let g ∈ L1(Fd). Then f ∗ g ∈ Lp(Fd) and ‖Tg‖Lp(Fd,dx) ≤ C2 ‖g‖L1(Fd,dx) , where C1 = C2 = ‖f‖Lp(Fd,dx). Riesz-Thorin interpolation yields ‖Tg‖Ls(Fd,dx) ≤ Cθ1C1−θ2 ‖g‖Lr(Fd,dx) = ‖f‖Lp(Fd,dx) ‖g‖Lr(Fd,dx) for any θ ∈ (0, 1), where r and s are given by 1 r = 1− θ 1 + θ p′ = 1− θ p and 1 s = 1− θ p + θ ∞ = 1− θ p . Let θ = 1 − ps . Note that ps > 0, so 1 − ps < 1. If 1s = 1p + 1r − 1, we have 0 < 1− 1r = 1p − 1s , so p < s and 0 < 1− ps . Thus we have ‖f ∗ g‖Ls(Fd,dx) ≤ ‖f‖Lp(Fd,dx) ‖g‖Lr(Fd,dx) with 1r = 1− 1− p s p = 1− 1p + 1s , so 1p + 1r = 1s + 1, as required. It remains to show the bounds (i) and (ii). Proof of (i): 13 Let f ∈ Lp(Fd), and g ∈ Lp′(Fd). Then |f ∗ g| = | ∫ f(x− y)g(y)dy| ≤ ∫ |f(x− y)||g(y)|dy ≤ ‖f‖Lp(Fd,dx) ‖g‖Lp′ (Fd,dx) (1.8) where (1.8) is from Hölder’s inequality. It follows that also ‖Tg‖L∞(Fd,dx) ‖f ∗ g‖L∞(Fd,dx) ≤ ‖f‖Lp(Fd,dx) ‖g‖Lp′ (Fd,dx) . Proof of (ii): Let f ∈ Lp(Fd) and g ∈ L1(Fd). We have ‖f ∗ g‖p Lp(Fd,dx) = ∫ | ∫ f(y)g(x− y)dy|pdx, and | ∫ f(y)g(x− y)dy| ≤ ∫ |f(y)| 1p′ |f(y)| 1p ||g(x− y)|dy ≤ ( ∫ |f(y)|dy) 1p′ ( ∫ |f(y)||g(x− y)|p) 1p (Hölder) = ‖f‖ 1 p′ L1(Fd,dx) ( ∫ |f(y)||g(x− y)|pdy) 1p so by Fubini (which amounts to changing the order of summation), we have ‖Tg‖p Lp(Fd,dx) = ‖f ∗ g‖ p Lp(Fd,dx) ≤ ∫ Fd ‖f‖ p p′ L1 ∫ Fd |f(y)||g(x− y)|pdydx = ‖f‖ p p′ L1(Fd,dx) ∫ ∫ |f(y)||g(x− y)|pdydx = ‖f‖ p p′ L1(Fd,dx) ∫ |f(y)| (∫ |g(x− y)|pdx ) dy ≤ ‖f‖ p p′ L1(Fd,dx) ‖f‖L1(Fd,dx) ‖g‖ p Lp(Fd,dx) and ‖f ∗ g‖Lp(Fd,dx) ≤ ‖f‖ 1 p′ L1(Fd,dx) ‖f‖ 1 p L1(Fd,dx) ‖g‖Lp(Fd,dx = ‖f‖L1(Fd,dx) ‖g‖Lp(Fd,dx), as required. This completes the proof. 14 Chapter 2 Progress Thus Far 2.1 Some Trivial Estimates A simple application of the results in Section 1.3 yields certain bounds immedi- ately, for any choice of the subset E. When r = ∞ it follows from |χ(x)| = 1 that ‖(fdσ)̌ ‖L∞(Fd∗,dσ) = maxξ ∣∣∣∣∣ 1|S|∑ x∈S χ(ξ · x)f(x) ∣∣∣∣∣ ≤ ‖f‖L1(Fd,dξ) so we have R(1 → ∞) for any subset of Fd. When r = 2, we use Plancherel to calculate ‖f‖L2(Fd,dx) = ‖f̂‖L2(Fd∗,dξ) =  1 |F|d ∑ ξ∈S |f̂ |2  12 = ( |S| |F|d ) 1 2  1 |S| ∑ ξ∈S |f̂ |2  12 = ( |S| |F|d ) 1 2 ‖f̂‖L2(S,dσ). From our nesting result in Theorem 1.3.2, this yields ‖f̂‖Lp′ (S,dσ) ≤ ‖f̂‖L2(S,dσ) = ( |F|d |S| ) 1 2 ‖f‖L2(Fd,dx) 15 for all p′ ≤ 2, which corresponds to 2 ≤ p ≤ ∞. Hence we have the bound R(p→ 2) ≤ ( |F|d |S| ) 1 2 for every 2 ≤ p ≤ ∞. Using f(x) ≡ 1 as a test function, we moreover see that R(p→ 2) = ( |F|d |S| ) 1 2 for all such p. Next, suppose 1 ≤ p1 ≤ p2 ≤ ∞, and fix 1 ≤ r ≤ ∞. Let C be such that ‖(gdσ)̌ ‖Lr(Fd,dx) ≤ C‖g‖Lp1 (S,dσ). Then from the nesting of Lp spaces with respect to a normalized measure, we have ‖(gdσ)̌ ‖Lr(Fd,dx) ≤ C‖g‖Lp2 (S,dσ) so R(p2 → r) ≤ R(p1 → r), for all ≤ p1 ≤ p2 ≤ ∞. (2.1) Similarly, letting 1 ≤ r1 ≤ r2, the monotonicity of Lp(Fd, dx) norms yields R(p→ r2) ≤ R(p→ r1), for all 1 ≤ p ≤ ∞. (2.2) Once a restriction bound is established, we can apply Riesz-Thorin interpo- lation (Theorem 1.3.3) with the trivial bound R(1→∞) . 1 in order to obtain a set of acceptable exponents. This strategy combined with the above mono- tonicity result establishes a range of exponents for which the extension estimate holds. 2.2 Necessary Conditions Let S ⊂ (Fd, dx) be an algebraic variety of size |S| ∼ qk for some 0 < k < d (in finite fields, we think of this as the dimension of S in Fd). When r ≥ 2, Mockenhaupt and Tao show that R(p→ r) . 1 can hold only if r ≥ 2d k , and (2.3) r ≥ dp ′ k = dp k(p− 1) . (2.4) Proof. We will use the trivial result (2.1). Let r1 ≤ r2. Then r′2 ≤ r′1, so the 16 nesting of normalized Lp spaces gives ‖g‖ Lr ′ 2 (Fd,dξ) ≤ ‖g‖ Lr ′ 1 (Fd,dξ) , that is, ( 1 |F|d ∫ |g|r′2 ) 1 r′2 ≤ ( 1 |F|d ∫ |g|r′1 ) 1 r′1 (∫ |g|r′2 ) 1 r′2 ≤|F|d( 1 r′2 − 1 r′1 ) (∫ |g|r′1 ) 1 r′1 =|F|d( 1r1− 1r2 ) (∫ |g|r′1 ) 1 r′1 . which implies R(p→ r1) ≤ qd( 1 r1 − 1 r2 ) R(p→ r2), for all 1 ≤ r1 ≤ r2 ≤ ∞. (2.5) Lastly we use our earlier calculation that R(p→ 2) = ( qd |S| ) 1 2 , when 2 ≤ p ≤ ∞. (2.6) If r ≥ 2, taking r1 = 2 and r2 = r in (2.5) and moving the constant gives R(p→ r) ≥ q−d( 12− 1r )R(p→ 2). From (2.6), this yields q−d( 1 2 − 1 r )R(p→ 2) ≥ q−d( 12− 1r )R(∞→ 2) = q−d( 1 2 − 1 r ) ( qd |S| ) 1 2 Therefore R(p→ r) ≥ q−d( 12− 1r )+ d2 |S|− 12 = q dr |S|− 12 . By assumption |S| ∼ qk, so we have R(p→ r) ≥ q dr− k2 but we require R(p→ r) ≤ C(p, r, d), so we must have dr− k2 ≤ 0, which rearranges to the necessary condition r ≥ 2dk , as claimed. 17 We obtain the second necessary condition by producing a potential coun- terexample. Let f(x) be the indicator function 1η, for some η ∈ S. Then (1.1) becomes ‖(1ηdσ)̌ ‖Lr(Fd,dx) ≤ R(p→ r) ‖1η‖Lp(S,dσ) . We calculate ‖1η‖Lp(S,dσ) = (∫ S 1pη ) 1 p = (∫ S 1η ) 1 p =  1 |S| ∑ ξ∈S 1η(ξ)  1p = |S|− 1p and ‖(1η )̌ ‖Lr(Fd,dx) = ∥∥∥∥∥∥ 1|S| ∑ ξ∈S 1η(ξ)χ(x · ξ) ∥∥∥∥∥∥ Lr(Fd,dx) = ∫ Fd  1 |S| ∑ ξ∈S 1η(ξ)χ(x · ξ) r dx  1 r = 1 |S| ∑ x∈Fd ∑ ξ∈S 1η(ξ)χ(x · ξ) r 1 r = 1 |S| ∑ x∈Fd (χ(x · η))r  1r = 1 |S| ∑ x∈Fd (χ(r(x · η)))  1r = 1 |S|q d r 18 In order for the bound (1.1) to hold for 1η we must have |S|−1q dr ≤ R(p→ r)|S|− 1p , Using |S| ∼ qk, this becomes |S| 1p−1q dr ∼ q k−kpp + dr ≤ R(p→ r) so we require k−kpp + d r ≤ 0, that is r ≥ dp k(p− 1) = dp′ k . Mockenhaupt and Tao further refine the necessary conditions in the case where S contains some affine subspace. If V ⊂ S is an affine subspace of dimen- sion l (i.e. |V | ∼ ql), they show that the restriction estimate will fail unless r ≥ p′ d− l k − l . (2.7) Proof. To see this, set f(x) = 1V (x). Then (1.1) becomes ‖(1V dσ)̌ ‖Lr(Fd,dx) ≤ R(p→ r) ‖1V ‖Lp(S,dσ) , where ‖1V ‖Lp(S,dσ) = (∫ S (1V (x)) pdσ ) 1 p =  1 |S| ∑ ξ∈S (1V (ξ)) p  1p = |S|− 1p ∑ ξ∈S 1V (ξ)  1p = |S|− 1p ∑ ξ∈V 1  1p = |S|− 1p |V | 1p 19 ∼ |S|− 1p q lp . Since V is an l−dimensional subspace, we can form a basis {ei}li=1 ⊂ Fd, and ‖(1V dσ)̌ ‖Lr(Fd,dx) = ∥∥∥∥∥∥ 1|S| ∑ ξ∈S 1V (ξ)χ(x · ξ) ∥∥∥∥∥∥ Lr(Fd,dx) = ∫ Fd  1 |S| ∑ ξ∈S 1V (ξ)χ(x · ξ) r dx  1 r = |S|−1 ∑ x∈Fd ∑ ξ∈S 1V (ξ)χ(x · ξ) r 1 r = |S|−1 ∑ x∈Fd ∑ ξ∈V χ(x · ξ) r 1 r = |S|−1 ∑ x∈Fd ∑ ξ∈V χ(x1ξ1)χ(x2ξ2) . . . χ(xlξl) r 1 r = |S|−1 ∑ x∈Fd ∑ ξ1∈F χ(x1ξ1) r . . . ∑ ξl∈F χ(xlξl) r 1 r and each inner sum is equal to zero unless xi = 0, so = |S|−1  ∑ x∈Fd s.t. x1=...=xl=0 qrqr . . . qr︸ ︷︷ ︸ l times  1 r = |S|−1qlq d−lr . This yields the necessary estimate |S|−1qlq d−lr ≤ R(p → r)|S|−1 1p q lp . Apply- ing |S| ∼ qk, this rearranges to ql+ d−lr − lp−k+ kp . R(p→ r), so we require l + d− l r − l − k p + k ≤ 0, 20 i.e. r ≥ p(d− l) (1− p)(l − k) = p ′ d− l k − l . 2.3 Tomas-Stein Estimates In the Euclidean case, the earliest non-trivial bound for the sphere is due to Tomas and Stein. It is known to hold for all smooth hypersurfaces of non- vanishing curvature on Rd, when p = 2. Let d be the dimension of Euclidean space. It states that we have R(p→ r) . 1 for all p and r such that r ≥ 2d+ 2 d− 1 and r ≥ p(d+ 1) (p− 1)(d− 1) . When p = 2, this gives the best possible estimate on the sphere. Tomas and Stein prove this by first rephrasing the problem in term of the Fourier transform on the surface measure dσ. For any finite measure µ, Tomas and Stein show that the estimate ∥∥∥f̂µ∥∥∥ Lp ≤ C ‖f‖L2(µ) for all f ∈ L2(µ) is equivalent to the condition ‖µ̂ ∗ h‖Lp ≤ C2 ‖h‖Lp′ for every Schwartz function h. They then prove the bound on the surface measure by applying a dyadic decomposition, and interpolating between bounds on the Bochner-Riesz kernel (which is essentially (dσ)̌ bounded away from one). In the finite setting, we have no such dyadic decomposition, but analogous principles of restating the problem in terms of (dσ)̌ and considering the Bochner-Riesz kernel can be put to use. Adapting the Euclidean approach in this way, Mockenhaupt and Tao prove that the Tomas-Stein estimate holds in the two-dimensional finite field case, when the subset is a paraboloid. The finite field Tomas-Stein type approach uses the orthogonality of T and T ∗ (see Remark 1) to express the desired estimate in a form that deals with (dσ)̌ . If we remove the origin from (dσ)̌ , it is often possible to obtain good decay estimates on its size. We define the Bochner-Riesz kernel K(x), which is (dσ)̌ with the origin removed. 2.3.1 Tomas-Stein type argument for even exponents By analyzing decay of (dσ)̌ away from the origin, Mockenhaupt and Tao obtain a set of sufficient conditions for restriction estimates from 2 to r, where r is even. The method employed relies on an analysis of the surface measure dσ, so gives 21 some of the flavour of the Tomas-Stein type method for exponents, although no specific restriction estimates are obtained here. Theorem 2.3.1 (Sufficient conditions for even exponents). Let r = 2m, m an integer, and let S ⊂ Fd. Suppose that for every ξ ∈ Fd∗, the size of {{ξi}mi=1 ⊂ S : ξ1 + ξ2 + . . .+ ξm = ξ} is bounded by A. Then we have the bound R(2→ 2m) ≤ A 12m |F | d2m |S|− 12 Proof. We want to show ‖(fdσ)̌ ‖L2m(Fd,dx) ≤ (A 1 2m |F | d2m |S|− 12 )‖f‖L2(S,dσ) Raising to the 2m, this becomes ‖(fdσ)̌ ‖2mL2m(Fd,dx) ≤ (A|F |d|S|−k)‖f‖2mL2(S,dσ) We use Plancherel and the intertwining of Fourier transforms and convolutions to calculate ‖(fdσ)̌ ‖2mL2m(Fd,dx) = ∫ ((fdσ)̌ )2mdx = ∫ (((fdσ)̌ )m)2 dx = ‖((fdσ)̌ )m‖2L2(Fd,dx) = ∥∥∥ ̂((fdσ)̌ )m∥∥∥2 L2(Fd∗,dx) 22 and by repeated application of (1.7) to ̂((fdσ)̌ )m, = ∥∥∥∥∥∥(̂fdσ)̌ ∗ . . . ∗ (̂fdσ)̌︸ ︷︷ ︸ m times ∥∥∥∥∥∥ 2 L2(Fd∗,dx) = ‖(fdσ) ∗ . . . ∗ (fdσ)‖2L2(Fd∗,dx) = ∫ Fd∗ ((fdσ ∗ . . . ∗ fdσ)(ξ))2dξ By Cauchy-Schwartz we have fdσ ∗ fdσ = (∫ f(ξ − η)dσ(ξ − η)f(η)dσ(η) )2 = (∫ f(ξ − η)(dσ(ξ − η)) 12 f(η)(dσ(η)) 12 (dσ(ξ − η)dσ(η)) 12 )2 ≤ ∫ f2(ξ − η)dσ(ξ − η)f2(η)dσ(η) ∫ dσ(ξ − η)dσ(η)) = (f2dσ ∗ f2dσ)(dσ ∗ dσ) which generalizes to (fdσ ∗ . . . ∗ fdσ)(ξ)(fdσ ∗ . . . ∗ fdσ)(ξ) ≤ (dσ ∗ . . . ∗ dσ)(ξ)(f2dσ ∗ . . . ∗ f2dσ)(ξ). Now dσ(ξ) := q d |S|1S(ξ), so dσ ∗ dσ(ξ) = ∫ Fd∗ dσ(η)dσ(ξ − η) = 1 qd ∑ η∈Fd qd |S|1S(η) qd |S|1S(ξ − η) = qd|S|−2 ∑ ξ∈Fd,η∈S ξη∈S 1 = qd|S|−2 ∑ ξ1,ξ2∈S ξ1+ξ2=ξ 1, and we can show inductively that dσ ∗ . . . ∗ dσ︸ ︷︷ ︸ k copies = qd|S|−k ∑ ξ1,...,ξk∈S ξ1+...+ξk=ξ 1. 23 Now by assumption ∑ ξ1,...,ξk∈S ξ1+...+ξk=ξ 1 ≤ A for all ξ ∈ Fd, so (fdσ ∗ . . . ∗ fdσ)(ξ))2 ≤ A|F |d|S|−k(f2dσ ∗ . . . ∗ f2dσ)(ξ) and ‖(fdσ)̌ ‖2mL2m(Fd,dx) ≤ A|F |d|S|−k ∫ Fd∗ (f2dσ ∗ . . . ∗ f2dσ)(ξ)dξ = A|F |d|S|−k‖f‖2mL2(S,dσ), as required. Note that this does not provide a restriction result as is, because the constant depends on the base field F. However, it does give us a nice sufficient condition for proving estimates of the form R(2 → 2m). If we can bound the size of S together with the number of solutions to ξ = ξ1 + . . .+ ξk in S such that the size of the field is canceled out in the constant, we have our result. Such a method is used by Koh and Shen in [10] to prove the R(2→ 4) estimate for certain cones in 3−dimensions (2.3.2). We use this same approach to extend the (2 → 4) result to non-degenerate cones in F4 (Theorem 3.0.1). 2.3.2 A Tomas-Stein type estimate for the cone In [10], Koh and Shen obtain the result R(2 → 4) for cones in 3−dimensional finite fields which do not contain any 2−dimensional affine spaces. They present two methods of proof — a geometric argument based on Section 2.3.1, and a Tomas-Stein type argument. The latter provides a good example for how such arguments work, so we detail it below: Theorem 2.3.2. Let P (x) ∈ F[x1, x2, x3] be a homogeneous polynomial such that H := {x ∈ F3 : P (x) = 0} does not contain any planes passing through the origin, and |H| ∼ q2. Then we have R(2→ 4) . 1. Proof. We will make use of the bound |(dσ)̌ (m)| . q−1, for all m 6= 0. (2.8) This was proved by Koh and Shen in [10] via Lemma 3.1.1, and an incidence 24 argument. Define the dual extension and restriction operators T ∗ : Lp(H, dσ)→ Lr(F3, dx);T ∗f := (fdσ)̌ and T : Lr ′ (F3, dx)→ Lp′(H, dσ);Tg := ĝ∣∣ H . From Remark 1, we have T ∗T (g) = g ∗ (dσ)̌ for all g on (F3, dx). We want to show ‖f̂‖2L2(H,dσ) . ‖f‖2L 43 (F3,dm). By orthogonality and Hölder’s inequality we have ‖f̂‖2L2(H,dσ) = ∫ H f̂ ¯̂ fdσ = ∫ F3 f̂ ∣∣ H ¯̂ f ∣∣ H dσ =< Tf, Tf >L2(H,dσ)=< T ∗Tf, f >L2(F3,dm) =< f ∗ (dσ)̌ , f >L2(F3,dx)≤ ‖f ∗ (dσ)̌ ‖L4(F3,dx)‖f‖L 4 3 (F3,dx) So it suffices to show ‖f ∗ (dσ)̌ ‖L4(F3,dx) . ‖f‖L 43 (F3,dx). This is where the analysis of (dσ)̌ comes in. We introduce the Bochner-Riesz kernel: Let K(x) := (dσ)̌ (x) − δ0(x), which is just (dσ)̌ with the origin set to 0, since (dσ)̌ (0) = 1 |H| ∑ x∈H χ(0, 0, 0) = 1. So ‖f ∗ (dσ)̌ ‖L4(F3,dx) = ‖f ∗ (K + δ0)‖L4(F3,dx) ≤ ‖f ∗K‖L4(F3,dx) + ‖f ∗ δ0‖L4(F3,dx) for all x For the second term, we can evaluate (f∗δ0)(x) = ∫ F3 f(y)δ0(x− y)dy = ∑ y∈F3 f(y)δ0(x− y) = f(x), so by Theorem 1.3.2, 25 ‖f ∗ δ0‖L4(F3,dx) = ‖f‖L4(F3,dx) ≤ ‖f‖L 43 (F3,dx) It remains to establish a bound on ‖f ∗ K‖L4(F3,dx). We obtain the necessary bound ‖f ∗K‖L4(F3,dx) . ‖f‖L 43 (F3,dx) by interpolating the following two bounds: (I) ‖f ∗K‖L4(F3,dx) . q‖f‖L2(F3,dx) (II) ‖f ∗K‖L∞(F3,dx) . q−1‖f‖L1(F3,dx) Setting θ = 12 in Riesz-Thorin interpolation then yields the bound for p and r such that 1 p = 1− 12 2 + 1 2 1 = 3 4 1 r = 1− 12 2 + 1 2 ∞ = 1 4 , with constant q 1 2 q− 1 2 = 1, as required. To complete the proof, we show the bounds (I) and (II). (I) : We have ‖f ∗K‖L2(F3,dx) = ‖ ̂(f ∗K)‖L2(F3,dx) (Plancherel) = ‖f̂ K̂‖L2(F3,dx) (by (1.7)) ≤ ‖K̂‖L∞(F3,dx)‖f̂‖L2(F3,dx) = ‖K̂‖L∞(F3,dx)‖f‖L2(F3,dx) and K̂(x) = dσ(x)− δ̂0(x) = q3|H|1H(x)− 1 . q for all x, since |H| ∼ q2, so ‖f ∗K‖L2(F3,dx) . q‖f‖L2(F3,dx). (II) : Applying Young’s inequality with r =∞, we have ‖f ∗K‖L∞(F3,dx) ≤ ‖f‖Lp(F3,dx)‖K‖Lp′ (F3,dx) in particular, ≤ ‖f‖L1‖K‖L∞ , and for all x ∈ F3, we have ‖K‖L∞ ≤ |(dσ)̌ (x)| . 1q , so we have ‖f ∗K‖L∞ . q−1‖f‖L1 , as required. 26 2.4 Dyadic Pigeonholing and Incidence Arguments Many of the known results for finite fields rely on arguments concerning the arithmetic structure of S, and bounds on the incidence of S with other subsets of Fd. We outline the reasoning in these arguments here, which can also be found B.J. Green’s lecture notes [4]. When r = 4, we can reduce the search for bound of the form (1.1) to a analysis of the structure of the subset S, and its incidence with other sets. We begin by obtaining an expression for the L4 norm of (fdσ)̌ in terms of additive quadruples, when f is a characteristic function. Lemma 2.4.1. Let E be any subset of the set S ⊂ Fd on which we are restricting f̂ . Then ‖(1Edσ)̌ ‖4L4(Fd,dξ) = qd |S|4 ∑ (ξ1,ξ2,ξ3,ξ4)∈E4 ξ1+ξ2=ξ3+ξ4 1 (2.9) Proof. We calculate ‖(1Edσ)̌ ‖4L4(Fd,dξ) = ∫ | ∫ χ(x · ξ)1E(ξ)dσ(ξ)|4dx = ∫  1 |S| ∑ ξ∈Fd∗ χ(x · ξ)1E(ξ) 4 dx = 1 |S|4 ∑ x∈Fd ∑ ξ∈Fd∗ χ(x · ξ)1E(ξ) 4 = 1 |S|4 ∑ x∈Fd  ∑ ξ1,ξ2,ξ3,ξ4∈E χ(x · ξ1)χ(x · ξ2)χ(x · ξ3)χ(x · ξ4)  27 changing variables and applying additivity, = 1 |S|4 ∑ x∈Fd  ∑ ξ1,ξ2,ξ3,ξ4∈E χ(x · (ξ1 + ξ2 − ξ3 − ξ4))  = 1 |S|4 ∑ ξ1,ξ2,ξ3,ξ4∈E ∑ x∈Fd χ(x · (ξ1 + ξ2 − ξ3 − ξ4))  by Lemma 1.2.1, = 1 |S|4 ∑ (ξ1,ξ2,ξ3,ξ4)∈E4 ξ1+ξ2=ξ3+ξ4 ∑ x∈Fd 1 = qd |S|2 ∑ (ξ1,ξ2,ξ3,ξ4)∈E4 ξ1+ξ2=ξ3+ξ4 1 It is also straightforward to calculate ‖1E‖4L2(S,dσ) = ( |E| |S| )2 . This implies that whenever we have the restriction result R(2 → 4) on S ⊂ Fd, we must have ∑ (ξ1,ξ2,ξ3,ξ4)∈E4 ξ1+ξ2=ξ3+ξ4 1 . |S| 2|E|2 |F|d . (2.10) For every E ⊂ S. This hints at the connection between restriction estimates over a subset S of a finite field and the arithmetic structure of the subset. Indeed, if we can bound the number additive quadruples in (2.10) for any subset of S, this line of argument can be further extended to arrive at new restriction results when r = 4. We arrive at these by incorporating the bound in (2.10) into a dyadic pigeonholing argument which allows us to consider only characteristic functions of subsets E, instead of more general functions on S. 28 2.4.1 The dyadic pigeonholing argument Let f be a bounded positive-valued function, supported on S. Without loss of generality, we may assume that ‖f‖L∞ = 1. We divide the range of f into dyadic shells, by defining Ej to be the set of all x such that 2 −j−1 < f(x) < 2−j , for every j ∈ N. Let fj := f · 1Ej , so that f is partitioned over these dyadic shells. By an argument similar to (2.9), for each fj we have ‖(fjdσ)̌ ‖44 = |F|d |S|4 ∑ (ξ1,ξ2,ξ3,ξ4)∈E4 ξ1+ξ2=ξ3+ξ4 f(ξ1)f(ξ2)f(ξ3)f(ξ4) ≤ |F| d |S|4 ∑ (ξ1,ξ2,ξ3,ξ4)∈E4 ξ1+ξ2=ξ3+ξ4 1 If we have good enough bound AEj on the sum of additive quadruples in terms of |Ej |, we can bound each ‖(fjdσ)̌ ‖4 by some Lp norm of f . Further, if we consider the tail of f , given by g := f · 1{x:f(x)<2−m}, we have the bound ‖(gdσ)̌ ‖44 = |F|d |S|4 ∑ (ξ1,ξ2,ξ3,ξ4)∈S4 ξ1+ξ2=ξ3+ξ4 g(ξ1)g(ξ2)g(ξ3)g(ξ4) ≤ A(∪∞j=mEj) |F|d |S|4 2 −4m, for large enough m. Since ‖f‖L∞ = 1, a good enough bound AE will allow us to bound ‖(gdσ)̌ ‖4 by the same Lp norm as above. Summing over this and the bound for the remaining fj ’s yields a restriction estimate for some R(p → 4), where p depends on the bound AE . An application of this method to the paraboloid in three dimensions is detailed by B.J.Green in [4], and is applied by Mockenhaupt and Tao in [17], and Iosevich and Koh in [6]. For an application to the sphere, see [7]. The key to proving exponents of this type becomes obtaining results on the arithmetic structure of S. 29 2.5 Summary of Existing Results In Euclidean space, the restriction problem is traditionally considered with re- spect to hypersurfaces with non-vanishing Gaussian curvature — in particular the sphere, cones, and paraboloids. Each of these has been studied to some extent over finite fields. In their introductory paper, Mockenhaupt and Tao fo- cus primarily on paraboloids and cones in low dimensions, and provide some counterexamples which serve as necessary conditions, as in (2.4) [17]. When S is given by the curve γ(t) := (t, t2, . . . , td), they use Theorem 2.3.1 to show R(p→ r) . 1 for every p, r such that r ≥ 2d, dp′. The necessary conditions in (2.4) show that this estimate is sharp. This yields R(2 → 2d) as the best estimate possible for the curve. 2.5.1 Paraboloids In the case of a paraboloid P , Mockenhaupt and Tao note that Theorem 2.3.1 proves R(2 → 4) . 1 in all dimensions. We always have |P | ∼ qd−1, so (2.4) gives us the necessary conditions r ≥ 2dd−1 , dp ′ d−1 .When d = 2, the authors show that these are also sufficient, so the sharpest estimate is R(2→ 4) . 1. If d = 3, they restrict themselves to finite fields F in which −1 is non-square (i.e. char(F) ≡ 3 mod 4). Without this specification P may contain lines, in which case (2.7) gives us the more restrictive necessary condition r ≥ 3p′. If P does not contain any lines, they conjecture that the conditions from (2.4) are necessary and sufficient. In this case (2.3) and (2.4) with d = 3 yields{ r ≥ 3p′2 , 3 } , so the conjectured sharp estimate is R(2→ 3). This is not proved, but they improve upon the 2 to 4 estimate in both directions, by showing bound- edness for R(2 → 185 + ) ∀  > 0 and R(85 → 4), up to a logarithmic factor of q. In 2008, Iosevich and Koh showed that when d ≥ 3 is odd and r = 4, the Tomas-Stein exponent is the best possible [6]. The authors obtain improved suf- ficient conditions for arbitrary finite fields with d ≥ 4, d even, up to a logarithmic factor of q. They further improve on the 2→ 4 estimate in both directions when d ≥ 7 is odd, with −1 non-square, again up a logarithmic factor. By applying a bilinear estimate in the stead of the dyadic argument, Lewko and Lewko obtain this exponent without the logarithmic factor in [11]. See Table 2.1 below for the specific results. 30 2.5.2 Quadratic Surfaces We can generalize the paraboloid to other non-degenerate quadratic surfaces. For x ∈ Fd with char(F ) > 2, let Q(x) be a homogeneous polynomial of degree 2, expressible as Q(x1, . . . , xd) = ∑d i,j=1 aijxixj , where aij = aji. If the matrix {aij} is non-singular, we call Q(x) a non-degenerate quadratic form. For any nonzero j in F, the non-degenerate quadratic surface Qj is given by Qj = {x ∈ Fd : Q(x1, . . . , xd) = j}. Using Gauss and Kloosterman sums (1.3 and 1.4) to derive incidence theorems for a Tomas-Stein type argument, Iosevich and Koh prove R(2 → 4) . 1 for any non-degenerate quadratic surface in Fd, d ≥ 2, and R(2 → r) . 1 for all r ≥ 2d+2d−1 . These are the only results on general quadratic forms to date [5]. 2.5.3 Spheres Another important special case of quadratic surfaces occurs when we take Q(x) = ‖x‖2. This is the finite field analog of the sphere. The above results for quadratic surfaces are valid here, and when d = 2 we already have the sharp estimate R(2→ 4). When d ≥ 3 is odd, Iosevich and Koh find that the Tomas-Stein exponent is the best possible, in [7]. When d ≥ 4 is even, they show R(p → 4) is bounded by a logarithmic factor of q whenever p ≥ 12d−89d−12 , which is an improvement upon the Tomas-Stein exponent p ≥ 4d−43d−5 . 2.5.4 Cones When j is equal to 0, Qj becomes the solution-set to a quadratic homogeneous variety, which is the finite field analog to the Euclidean cone. Koh and Shen study restriction estimates for these varieties in two and three dimensions ([9] and [10]). When d = 2 and the characteristic of F is not equal to two, they show that the (2 → 4) holds on surfaces defined by homogeneous varieties of size O(q), provided the variety has no linear factor. Further, they show that interpolation with this bound and application of Theorem 1.3.2 yields the best range of exponents possible in this case. In [10], an application of Theorem 2.3.1 yields the same estimate for three-dimensional homogeneous varieties with size O(q2) which contain no planes passing through the origin. A counterexample from [17] shows that this is the best estimate possible for homogeneous varieties of this type. 31 Known estimates Necessary conditions Curves [17] General d 2→ 2d∗ r ≥ 2d, γ(t) := (t, t2, · · · , td) r ≥ dp′ Non-Degenerate Quadratic Surfaces [5] General d 2→ 4 r ≥ 4, 2→ 2d+2d−1 r ≥ 2pp−1 Paraboloids [6][11][17] General d 2→ 4 r ≥ 2dd−1 , dp ′ d−1 d = 2 2→ 4∗ r ≥ 4, 2p′ d = 3, −1 nonsquare 2→ 18 5 +  r ≥ 3 8 5 → 4 r ≥ 3p ′ 2 d = 3, −1 square 2→ 4 (Best p given r = 4) r ≥ 3, 2p′ d ≥ 3, odd, −1 nonsquare 85 → 4 d ≥ 4, even 4d 3d−2 → 4 r ≥ 2dd−1 2→ 2d2 d2−2d+2 r ≥ p(d+2)(p−1)d d ≥ 7,−1 nonsquare, 4d3d−2 → 4 q = ℘l, 4 - l(d− 1) 2→ 2d2 d2−2d+2 Spheres [7][17] d = 2 2→ 4 r ≥ 4, 2p′ d ≥ 3, odd 2→ 2d+2d−1 r ≥ 2dd−1 , dp ′ d−1 d ≥ 4, even 12d−8 9d−12 → 4 r ≥ 2dd−1 , 2→ 2d+2d−1 r ≥ dp ′ d−1 Cones [9][10] d = 2 2→ 4∗ r ≥ 4, 2p′ d = 3 2→ 4∗ r ≥ 4,|S| ∼ q2, no planes r ≥ 2p′ Note: Results marked with an asterisk are optimal, in the sense that the value of r is the smallest possible, and p is the minimal possible value for this r. Table 2.1: Summary of Existing Results for p→ r. 32 Chapter 3 Restriction Estimates for Cones over Finite Fields In [10] Koh and Shen arrive at the (2 → 4) estimate for the three dimensional cone by proving incidence results on homogeneous varieties and applying them to Mockenhaupt and Tao’s sufficient condition for even exponents (Theorem 1.3.2). In what follows we adapt their techniques to homogeneous in general dimensions. Theorem 3.0.1. Let P (x) be a d-variate polynomial over F, and H = {x ∈ Fd : P (x) = 0}. Suppose that |H| ∼ qd−1, and that the homogeneous variety H is expressible as a disjoint (except at the origin) union of k−dimensional planes in Fd for a fixed k ≤ d − 2, and does not contain planes of any dimension greater than k. Then we have the following restriction estimate on H: R(2→ 4) . 1. Theorem 3.0.2. The restriction estimate R(2→ 4) . 1 holds for any homoge- neous variety H ∈ Fd, provided |H| ∼ qd−1 and H does not contain a hyperplane passing through the origin. 3.1 Preliminary Lemmas In the proof of Theorem 3.0.1, we will follow the methods of [9], generalized to the higher-dimensional setting. We will require the following lemmas. 33 Lemma 3.1.1 (Schwartz-Zippel). If P (x) is a d-variate polynomial over F of degree k 6= 0, then |{x ∈ Fd : P (x) = 0}| ≤ kqd−1. Proof. This lemma was proven independently by Schwartz [15], Zippel [21] and Demillo-Lipton [1] from a probabilistic perspective, using inductive methods. We will follow a more recent proof by Dana Moshkovitz [13]. First note that when d = 1 this is just the factor theorem, and if k ≥ |F|, then the number of zeros is at most |F|d ≤ k|F|d−1, so the result is trivial. It suffices to consider d ≥ 2 and 1 ≤ k < |F|. We will reduce these to the case d = 1, by restricting the function to one-dimensional objects. Let P (x) ∈ F[x1, . . . , xd] be a nonzero polynomial of degree k 6= 0. We can decompose P (x) into P (x) = PH(x) + p(x), where PH(x) consists of all highest- degree monomials in P (x). Note that PH(x) is nonzero and homogeneous, of degree k. For now we assume that there exists some nonzero element y ∈ Fd such than PH(y) 6= 0 (this will be established in Lemma 3.1.2.) For such a y, we can partition the elements of Fd according to lines with direction y, so that Fd = unionsqx∈Fd−1{(x, xd) + ty|t ∈ F}, where xd is fixed. Note that there are |F|d−1 such lines. Now consider the restrictions of P to such lines. P (x + ty) is a kth-degree univariate polynomial in t. Note that the coefficient of tk is exactly the k-degree terms of P evaluated at y, i.e. PH(y). Since PH(y) 6= 0, P (x + ty) is nonzero. By the d = 1 case, the number of zeros of P restricted to a given such line is at most k. The total number of zeros on Fd is therefore at most k|F|d−1. Lemma 3.1.2. Let P (x) ∈ F[x1, . . . , xd] be a nonzero polynomial with degree 1 ≤ k < |F|. Then there exists some y ∈ Fd for which P (y) 6= 0. Moreover, if P is homogeneous, y is nonzero. Proof. Note that when d = 1, P has at most k < |F| zeros, so the lemma holds. Suppose by way of contradiction that the lemma does not hold for general d. Let m be the smallest value for which the lemma fails. Then there exists P ∈ F[x1, . . . , xm] such that P (x1, . . . , xm) = 0 for every (x1, . . . , xm) ∈ Fm. For a fixed a ∈ F, let Pa(x1, . . . , xm−1) := P (a, x1, . . . , xm−1). Since Pa is (m − 1)- variate, there exists y = (y1, . . . , ym−1) such that Pa(y) 6= 0. But then for ya := (a, y1, . . . , ym−1) ∈ Fm we have P (ya) = Pa(y) = 0, a contradiction. Hence the lemma holds for all d. Finally, if P is homogeneous, P (0) = 0. 34 Lemma 3.1.3. Fix k ≤ d − 1. Suppose H = {x ∈ Fd : P (x) = 0} is a homoge- neous variety, with P (x) 6≡ 0. Then for any k−dimensional plane ΠM 6⊂ H, we have |H ∩ΠM | . qk−1. Proof. Since ΠM is a k−dimensional plane in Fd, without loss of generality we may assume there exists k − dimensional vectors {Mj}dj=k+1, and M0 ∈ Fd such that ΠM = {M0 + (x1, . . . , xk,Mk+1 · (x1, x2, . . . , xk), . . . ,Md · (x1, x2, . . . , xk))}. Since every element of H ∩ ΠM is expressible in this form and also satisfies P (x) = 0, we have |H ∩ΠM | = |y ∈ Fk : P (M0 + (y1, . . . , yk,Mk+1 · y, . . . ,Md · y)) = 0|. Now P (M0 + (y1, . . . , yk,Mk+1 · y, . . . ,Md · y)) is a k−variate polynomial. Since ΠM 6⊂ H, it is not identically zero. The result therefore follows from Lemma 3.1.1. The proof of Theorem 3.0.1 will rely on the following lemma, the details of which can also be found in [17] and [5]. Lemma 3.1.4. Let V be any algebraic variety in Fd, d ≥ 2, with |V | ∼ qd−1. Suppose that for every ξ ∈ Fd\{(0, . . . , 0)},∑ (x,y)∈V×V :x+y=ξ 1 . qd−2. Then, we have R(2→ 4) . 1. Proof. We apply the bound on {(x, y) ∈ V × V : x + y = ξ} to Theorem 2.3.1, with m = 2. This gives R(2→ 4) . (qd−2) 14 q d4 |V |− 12 and |V | ∼ qd−1, so R(2→ 4) . q d−24 + d4− d−12 = q d−2+d−2d+2 4 = 1 35 so R(2→ 4) . 1, independent of the base field. 3.2 Proof of Theorems 3.0.1 and 3.0.2 3.2.1 Proof of Theorem 3.0.1 Let H be a homogeneous variety in Fd, expressible as a union of k−dimensional planes intersecting only at origin, for some fixed k ≤ d−2. Assume that |H| does not contain any planes of dimension greater than k, and |H| ∼ qd−1. Applying Lemma 3.1.4, it suffices to show that for every ξ ∈ Fd \ {(0, . . . , 0)}, we have∑ (x,y)∈H×H:x+y=ξ 1 . qd−2. Equivalently, we must show |{x ∈ Fd : P (x) = P (ξ−x) = 0}| . qd−2. It thus suffices to prove |H ∩ (H + ξ)| . qd−2, for all ξ 6= 0. Fix ξ ∈ Fd \ {(0, . . . , 0)}. By assumption, we have H = ∪nj=1Πj , where each Πj is a k−dimensional plane in Fd. Note that each plane has size ∼ qk, so we have n ∼ qd−k−1. Since |H ∩ (H + ξ)| ≤ n∑ j=1 |H ∩ (Πj + ξ)|, it will suffice to consider the sets H ∩ (Πj + ξ). If ξ ∈ Πj , then |H ∩ (Πj + ξ)| = |H ∩ Πj| ∼ qk. Since ξ 6= 0 and the planes are disjoint except at the origin, this occurs at most once. We have |H ∩ (H + ξ)| ≤ qk + qd−k−1|H ∩ (Πl + ξ)|, where Πl is one of the k-dimensional planes comprising H and not containing ξ. If ξ /∈ Πl, then Πl + ξ does not intersect the origin, and hence is distinct from each Πj comprising H = ∪nj=1Πj . Since 0 6∈ (Πl + ξ), if Πl + ξ was contained in H, H would contain the (k + 1)-dimensional plane spanned by 0 and Πl + ξ, a contradiction. Hence we have (Πl + ξ) 6⊂ H. From Lemma 3.1.3, it follows that |H ∩ (Πl + ξ)| . qk−1, 36 so finally |H ∩ (H + ξ)| ≤ qk + qd−k−1qk−1 . qd−2, as required. 3.2.2 Proof of Theorem 3.0.2 The main result will follow as a consequence of Theorem 3.0.1. Let H be a homo- geneous variety containing no hyperplanes, with H ∼ qd−1. We may decompose H into collections of planes of the same dimension, so that H = ∪k≤d−2Hk, where Hk = ∪jΠj , such that {Πj} are planes of dimension k, and Hk contains no planes of dimension greater than k. For all k 6= l, we have Hk ∩Hl = {(0, . . . , 0)}. We have ‖f‖L2(H,dσ) ≤ d−2∑ k=1 ‖f‖L2(Hk,dσ) (3.1) . d−2∑ k=1 ∥∥∥f̂∥∥∥ L4(Hk,dσ) . d−2∑ k=1 ∥∥∥f̂∥∥∥ L4(H,dσ) (3.2) . ∥∥∥f̂∥∥∥ L4(H,dσ) . Where (3.2) follows from Theorem 3.0.1. Hence we have the estimate R(2→ 4) on H. Remark 2. In general we would expect our estimate to worsen for more degen- erate varieties. Hence if we impose stricter conditions on the highest-dimension planes present in H, we expect that a sharper restriction estimate is possible. In particular, when d = 4 it is believed that a sharper estimate will hold in the case where H contains no 2−dimensional planes. 37 3.3 Possible Improvements If S ⊂ Fd has dimension k, Mockenhaupt and Tao proved the necessary conditions r ≥ 2d k , (3.3) r ≥ d k p′ (3.4) for R(p→ r) bounded by O(1). In the case of a 3−dimensional cone in F4, these yield the necessary conditions r ≥ 8/3 and p ≥ 3r3r−4 . The first of these bounds will be improved to r ≥ 3 in what follows, for general 3−dimensional cones in F4. If we set r = 3, the second bound yields the necessary condition p ≥ 9/5. Moreover, if S is known to contain a k−dimensional subspace, they prove the necessary condition r ≥ p′ d− l k − l . (3.5) Since every cone will contain 1−dimensional lines, in our case this implies r ≥ 32p′, so that p ≥ 2r2r−3 . When r = 4, we obtain a lower bound of 85 on p. On the other hand, in the case that our cone contains a 2−dimensional plane, we obtain r ≥ 2p′, which leads to p ≥ rr−2 , and when r = 4, this gives the necessary condition p ≥ 4. Hence the bound found in Theorem 3.0.2 gives the best possible range for p, in the case where H contains a 2−dimensional plane. If H only contains lines, (3.5) suggests it may be possible to reduced p to as low as 85 , when r = 4. As we will see in Theorem 3.4.1, for general 3−dimensional cones in F4 we have the lower bound r ≥ 3, an improvement upon (3.3). If H contains a 2−dimensional subspace, r = 3 implies p ≥ 3. If H has no such planes, we may have r = 3, p ≥ 2. 3.4 Upper Bounds on r for Specific Cones In the case of the general three dimensional cone, Mockenhaupt and Tao produced a counterexample to the bound (p → r), for all r < 4, and all p [17]. We adapt this counterexample to the four dimensional case. Theorem 3.4.1. Let S := {(x, y, z, w) ∈ F4 : x2 + y2 = zw}. (Note that by change of variables this is equivalent to the cone x2 + y2 + z2 − w2 = 0.) Then 38 for all r ≤ 3, 1 ≤ p ≤ ∞, R(p→ r) is unbounded. Proof. We consider f = 1X , where X is a subset of S given by X = {(x′, y′, z′, w′) : z′ is a square, 4z′w′ = x′2 + y′2}. Since S is a three dimensional cone, we have |X| ∼ |F|3. We can make the change of variables {x′ = tu, y′ = su, z′ = (t2 + s2)u,w′ = u} for all (x′, y′, z′, w′) on S, so we have 1̂X(x ′, y′, z′, w′) = 1̂X(tu, su, (t2 + s2)u, u) (3.6) = ∑ z′∈Q ∑ x′,y′F χ(tux′ + suy′ + (t2 + s2)uz′ + u(x′2 + y′2)/4z′), where Q is the set of non-zero squares in F, and χ is a non-trivial additive character of F. Rearranging and completing two squares, we have 1̂X(x ′, y′, z′, w′) = ∑ z′∈Q ∑ x′,y′∈F χ( u 4z′ (4tx′z′ + 4sy′z′ + 4(t2 + s2)z′2 + x′2 + y′2)) = ∑ z′∈Q ∑ x′,y′∈F χ( u 4z′ ((x′2 + 4tx′z′ + 4t2z′2) + (y′2 + 4sy′z′ + 4s2z′2))) = ∑ z′∈Q ∑ x′,y′∈F χ( u 4z′ ((x′ + 2tz′)2)χ( u 4z′ ((y′ + 2sz′)2) = ∑ z′∈Q [ ∑ x′∈F χ( u 4z′ ((x′ + 2tz′)2) ∑ y′∈F χ( u 4z′ ((y′ + 2sz′)2)] = ∑ z′∈Q [ ∑ ξ∈F χ(uξ2) ∑ η∈F χ(uη2)], where ξ = x ′+2tz′ 2 √ z′ , η = y ′+2sz′ 2 √ z′ . Finally, applying the identity from Theorem 1.4 yields |1̂X | = |Q||F| = O(|F|2). Since |X| ∼ |F|, we have |1̂X | ∼ |X| 23 . Thus for all 1 ≤ p ≤ ∞ we have ‖1̂X‖Lp′ & |1̂X | ∼ |X| 2 3 > |X|Lr′ , 39 for all r′ > 3/2, which corresponds to r < 3. We can follow much the same procedure for the four-dimensional cone of the form S′ := {(x, y, z, w) ∈ Fd : x2 − y2 = wz}. Making the change of variables {x = tu, y = tu, z = (t2 − s2)u,w = u} on S′ and considering the characteristic function of the set X ′ := {(x′, y′, z′, w′) : z′ is a square, 4z′w′ = x′2 − y′2} yields the above result. 40 Chapter 4 Conclusion The preceding work adds the bound 2→ 4 for general dimensional cones without hypersurfaces to the body of known results, and demonstrates r = 4 cannot be improved in this case. As we have seen from this discussion as well as Table 2.1, lines of investigation remain in several directions. Although Mockenhaupt and Tao completely solve the problem over the curve γ(t), there are no results on curves of any other form thus far. In the case of the paraboloid, Mockenhaupt and Tao conjecture that their bounds in three dimen- sions can be improved to 3→ 3 when −1 is a square, and 2→ 3 otherwise. They also expect that counterexamples to these bounds exist in higher dimensions. Consulting Table 2.1, we note that counterexamples and refined results need to be sought for each of the quadratic surfaces under consideration, when d is large. In the case of high-dimensional cones, we expect to obtain improvements upon the 2 to 4 bound when we restrict ourselves to cones of lower degeneracy. Over four-dimensional fields, if we assume that the homogeneous variety H does not contain any 2 or 3 dimensional planes passing through the origin, such an improvement should be possible. In in [6] and [7], Iosevich and Koh find incidence bounds via the bounds on Gauss sums found in Section 1.2. Applying these incidence results to a dyadic pigeonholing argument yields restriction results for certain cases of the paraboloid and sphere. Attempting these methods on the cone provides incidence bounds which are not sharp enough to lead to an improvement on the 2→ 4 result. In the Euclidean case, it is sometimes possible to improve upon restriction results by passing to a bilinear estimate. These consider bounds on a product of 41 two functions supported on non-overlapping subsets of S, so that (1.1) becomes ‖(f1dσ1)̌(f2dσ2)̌‖ ‖Lr(Fd,dξ) . ‖f1‖Lp(S1,dσ) ‖f2‖Lp(S2,dσ2) . Recently, Lewko and Lewko adapted this approach to the finite field setting in order to obtain bounds for paraboloids and spheres that were previously only known up to a logarithmic factor of q [11]. However, it is unclear precisely what bilinear estimates can imply about linear estimates over finite fields in the case of the cone. An improved understanding of finite bilinear restriction theorems, of the form obtained by Tao and Vargas for the Euclidean setting (see [19], [18]), would be very useful. A result linking linear and bilinear results for the finite cone may lead to progress on this problem. Considering functions supported over sections of the cone that do not share any common directions may allow us to improve our incidence theorems, and arrive at an improved restriction estimate. 42 Bibliography [1] R. A. Demillo and R. J. Lipton. A probabilistic remark on algebraic program testing. Information Processing Letters, 7(4):193 – 195, 1978. → pages 34 [2] Z. Dvir. On the size of Kakeya sets in finite fields. J. Amer. Math. Soc., 22:1093–1097, 2009. → pages 3 [3] G. Folland. Real analysis: modern techniques and their applications. Wiley-Interscience, Toronto, 2 edition. ISBN 0471317160. → pages 1 [4] B. Green. Restriction and kakeya phonomena. http://www.dpmms.cam.ac.uk/∼bjg23/rkp.html, 2003. → pages 27, 29 [5] A. Iosevich and D. Koh. Extension theorems for the Fourier transform associated with nondegenerate quadratic surfaces in vector spaces over finite fields. Illinois Journal of Mathematics, 52(2):611–628, 2008. → pages 31, 32, 35 [6] A. Iosevich and D. Koh. Extension theorems for paraboloids in the finite field setting. Mathematische Zeitschrift, 266(2):471–487, July 2009. → pages 29, 30, 32, 41 [7] A. Iosevich and D. Koh. Extension theorems for spheres in the finite field setting. Forum Mathematicum, 22(3):457–483, May 2010. → pages 29, 31, 32, 41 [8] Y. Katznelson. An introduction to harmonic analysis. Cambridge mathematical library. Cambridge University Press, 2004. ISBN 9780521543590. URL http://books.google.ca/books?id=gkpUE m5vvsC. → pages 12 [9] D. Koh and C. Shen. Sharp extension theorems and Falconer distance problems for algebraic curves in two dimensional vector spaces over finite fields. Arxiv preprint arXiv:1003.4240, 28(1), 2010. URL http://arxiv.org/abs/1003.4240. → pages 31, 32, 33 43 [10] D. Koh and C. Shen. Harmonic analysis related to homogeneous varieties in three dimensional vector space over finite fields. Arxiv preprint arXiv:1005.2644, pages 1–18, 2010. URL http://arxiv.org/abs/1005.2644. → pages 24, 31, 32, 33 [11] A. Lewko and M. Lewko. Restriction estimates for the paraboloid over finite fields. ArXiv e-prints, Sept. 2010. → pages 30, 32, 42 [12] R. Lidl and H. Niederreiter. Introduction to Finite Fields and Their Applications, volume 79. Cambridge University Press, Cambridge, 1 edition, July 1986. ISBN 0521307066. doi:10.2307/3618355. → pages 5 [13] D. Moshkovitz. An alternative proof of the schwartz-zippel lemma. Electronic Colloquium on Computational Complexity (ECCC), 17:96, 2010. → pages 34 [14] W. Rudin. Real and complex analysis. McGraw-Hill Book Co., New York, third edition, 1987. ISBN 0-07-054234-1. → pages 9 [15] J. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. Journal of the ACM, 27:701–717, 1980. → pages 34 [16] T. Tao. A sharp bilinear restriction estimate for paraboloids. Geometric And Functional Analysis, 13(6):1359–1384, Dec. 2003. → pages 2 [17] T. Tao and G. Mockenhaupt. Restriction and Kakeya phenomena for finite fields. Duke Mathematical Journal, 121(1):35–74, Jan. 2004. ISSN 0012-7094. → pages 29, 30, 31, 32, 35, 38 [18] T. Tao and A. Vargas. A bilinear approach to cone multipliers I. Restriction estimates. Geometric And Functional Analysis, 10(1):185–215, Apr. 2000. → pages 42 [19] T. Tao, A. Vargas, and L. Vega. A bilinear approach to the restriction and kakeya conjectures. Journal of the American Mathematical Society, 11(4): pp. 967–1000, 1998. ISSN 08940347. URL http://www.jstor.org/stable/2646145. → pages 42 [20] T. Wolff. A sharp bilinear cone restriction estimate. The Annals of Mathematics, 153(3):pp. 661–698, 2001. ISSN 0003486X. URL http://www.jstor.org/stable/2661365. → pages 2 [21] R. Zippel. Probabilistic algorithms for sparse polynomials. In In Proceedings of the International Symposium on Symbolic and Algebraic Computation, pages 216–226, 1979. → pages 34 44

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