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Trainor, Charlotte

University of British Columbia

The finite field Kakeya conjecture concerns the size of subsets of 𝔽ⁿ𝘲 that contain a line in each direction, and is a discrete analogue of a major open problem in harmonic analysis. In 2008, Dvir found an elegant solution to this conjecture using elementary properties of polynomials. His proof popularized the polynomial method, which has proved to be a powerful strategy to tackle problems in analysis and discrete mathematics. This dissertation concerns two main research problems emerging from these areas. In the first, we consider a variant of the Kakeya problem. Besicovitch-Rado-Kinney (BRK) sets in ℝⁿ contain a sphere of radius 𝘳, for each 𝘳 > 0. It is known that such sets have dimension 𝘯 from the work of Kolasa and Wolff. We consider a discrete version of this problem. We define BRK-type sets in 𝔽ⁿ𝘲, and establish lower bounds on the size of such sets using techniques introduced by Dvir’s proof of the finite field Kakeya conjecture. For our second main research problem, we study connections between hyperplanes and generalized polynomials in (ℤ/𝘱ᵏℤ)ⁿ. Let 𝘏ⁿ be the linear span of characteristic functions of hyperplanes in (ℤ/𝘱ᵏℤ)ⁿ. We establish new upper bounds on the dimension of 𝘏ⁿ over ℤ/𝘱ℤ, or equivalently, on the rank of point-hyperplane incidence matrices in (ℤ/𝘱ᵏℤ)ⁿ over ℤ/𝘱ℤ. Our proof is based on a variant of the polynomial method using binomial coefficients in ℤ/𝘱ᵏℤ as generalized polynomials. We also establish additional necessary conditions for a function on (ℤ/𝘱ᵏℤ)ⁿ to be an element of 𝘏ⁿ.

Text

2024-07-22

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/88668

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/