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Restriction theorems and Salem sets Hambrook, Kyle David


In the first part of this thesis, I prove the sharpness of the exponent range in the L² Fourier restriction theorem due to Mockenhaupt and Mitsis (with endpoint estimate due to Bak and Seeger) for measures on ℝ. The proof is based on a random Cantor-type construction of Salem sets due to Laba and Pramanik. The key new idea is to embed in the Salem set a small deterministic Cantor set that disrupts the restriction estimate for the natural measure on the Salem set but does not disrupt the measure's Fourier decay. In the second part of this thesis, I prove a lower bound on the Fourier dimension of Ε(ℚ,ψ,θ) = {x ∊ ℝ : ‖qx - θ‖ ≤ ψ(q) for infinitely many q ∊ ℚ}, where ℚ is an infinite subset of ℤ, Ψ : ℤ → (0,∞), and θ ∊ ℝ. This generalizes theorems of Kaufman and Bluhm and yields new explicit examples of Salem sets. I also prove a multi-dimensional analog of this result. I give applications of these results to metrical Diophantine approximation and determine the Hausdorff dimension of Ε(ℚ,ψ,θ) in new cases.

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