Explicit Salem sets and applications to metrical Diophantine approximation

2016/04/01 by Hambrook, Kyle
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1604.00411

Abstract

Let Q be an infinite subset of ℤ, let Ψ: ℤ → [0,∞) be positive on Q, and let θ∈ ℝ. Define E(Q,Ψ,θ) = \ x ∈ ℝ : ‖ q x - θ‖ ≤ Ψ(q) for infinitely many q ∈ Q \. We prove a lower bound on the Fourier dimension of E(Q,Ψ,θ). This generalizes theorems of Kaufman and Bluhm and yields new explicit examples of Salem sets. We give applications to metrical Diophantine approximation, including determining the Hausdorff dimension of E(Q,Ψ,θ) in new cases. We also prove a higher-dimensional analog of our result.

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