On the projections of Ahlfors regular sets in the plane

2024/10/09 by Tuomas Orponen, Orponen, Tuomas
Mathematics · Computer Science · #Mathematical Approximation and Integration #Optimization and Variational Analysis #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2410.06872

Abstract

This paper contains the following δ-discretised projection theorem for Ahlfors regular sets in the plane. For all C,ε> 0 and s ∈ [0,1], there exists κ> 0 such that the following holds for all δ> 0 small enough. Let ν be a Borel probability measure on S¹ satisfying ν(B(x,r)) ≤ Cr^ε for all x ∈ S¹ and r > 0. Let K ⊂ B(1) ⊂ ℝ² be Ahlfors s-regular with constant at most C. Then, there exists a vector θ∈ spt ν such that |π_θ(F)|_δ ≥ δ^{ε- s} for all F ⊂ K with |F|_δ ≥ δ^{κ- s}. Here π_θ(z) = θ⋅ z for z ∈ ℝ².

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