On the distance sets of Ahlfors–David regular sets

By Tuomas Orponen. 2015

49 citations recorded by the source corpus.

Abstract I prove that if ∅ ≠ K ⊂ R 2 is a compact s-Ahlfors–David regular set with s ≥ 1 , then dim p ⁡ D ( K ) = 1 , where D ( K ) : = { | x − y | : x , y ∈ K } is the distance set of K, and dim p stands for packing dimension. The same proof strategy applies to other problems of similar nature. For instance, one can show that if ∅ ≠ K ⊂ R 2 is a compact s-Ahlfors–David regular set with s ≥ 1 , then there exists a point x 0 ∈ K such that dim p ⁡ K ⋅ ( K − x 0 ) = 1 .

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9480267d4f82e1a0c83e6b57d1feb8c2431f3a21

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

New Bounds on the Dimensions of Planar Distance Sets (2018, methodology): “IMENSIONS OF PLANAR DISTANCE SETS 17 local pinned distance measures. We emphasize that this method to bound the dimension of (linear or nonlinear) projections from below goes back in various forms to [10,9,22], although the use of projected L2 norms (rather than projected entropies) was ﬁrst used in”
Assouad Dimension and Fractal Geometry (2020, background): “That said, since 2010 the problem has been receiving a lot of attention in the literature, and significant progress is being made, see for example, [252, 253, 225, 167, 124].”
Weighted restriction estimates and application to Falconer distance set problem (2018, background): “For instance, in two dimensions the full Falconer conjecture has been established by Orponen [20] under the additional assumption that the set considered is Ahlfors-David regular, with a slightly weaker conclusion that the upper Minkowski dimension of its distance set is equal to 1.”
Incidence Estimates for Well Spaced Tubes (2019, methodology): “ Falconerproblem In this section, we consider a distinct distances type problem for δ-balls in R2, which is related to the Falconer distance problem in R2. As we mentioned in the introduction, Orponen[16] and Keleti-Shmerkin[23][24][14] essentially solved the Falconer distance problem for sets that ”