Abstract
We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if Asubset {mathbb {R}}^2 is a Borel set of Hausdorff dimension s>1, then its distance set has Hausdorff dimension at least 37/54approx 0.685. Moreover, if sin (1,3/2], then outside of a set of exceptional y of Hausdorff dimension at most 1, the pinned distance set { |x-y|:xin A} has Hausdorff dimension ge tfrac{2}{3}s and packing dimension at least tfrac{1}{4}(1+s+sqrt{3s(2-s)}) ge 0.933. These estimates improve upon the existing ones by Bourgain, Wolff, Peres–Schlag and Iosevich–Liu for sets of Hausdorff dimension >1. Our proof uses a multi-scale decomposition of measures in which, unlike previous works, we are able to choose the scales subject to certain constrains. This leads to a combinatorial problem, which is a key new ingredient of our approach, and which we solve completely by optimizing certain variation of Lipschitz functions.
Highlights
In this article we prove new lower bounds on the dimensions of distance sets, which in particular greatly improve the best previously known estimates when dimH(A) = 1 + δ, δ > 0 small
The fact that one can improve upon Theorem 1.1 is based on the observation that for some sets A ⊂ R2 of Hausdorff dimension s > 1 for which the method of the Proof of Theorem 1.1 cannot give anything better than dimH(Δ(A)) ≥ 2s/3, the quantitative version of Wolff’s Theorem can give a much better bound
We derive a lower bound on box-counting numbers of pinned distance sets that will be crucial in the proofs of Theorems 1.2, 1.3 and 1.4
Summary
After this paper was made public, Liu [Liu18] posted a preprint extending Wolff’s result to pinned distance sets He shows that if A ⊂ R2 is a Borel set with dimH(A) > 4/3, Δx(A) has positive Lebesgue measure for some x ∈ A (with bounds on the dimension of the exceptional set). The fact that one can improve upon Theorem 1.1 (for the full distance set) is based on the observation that for some sets A ⊂ R2 of Hausdorff dimension s > 1 for which the method of the Proof of Theorem 1.1 cannot give anything better than dimH(Δ(A)) ≥ 2s/3, the quantitative version of Wolff’s Theorem can give a much better bound The fact that these two methods are based on totally different techniques and have different “enemies” that one must overcome, suggests that neither of them (or even in combination as we do here) provides a definitive line of attack on Falconer’s problem. Orponen for many useful discussions at the early stage of this project, and an anonymous referee for several suggestions that improved the paper, and in particular for suggesting a simplification of the statement and Proof of Proposition 3.12
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
