• Corpus ID: 245218865

# On the distance sets spanned by sets of dimension $d/2$ in $\mathbb{R}^d$

@inproceedings{Shmerkin2021OnTD,
title={On the distance sets spanned by sets of dimension \$d/2\$ in \$\mathbb\{R\}^d\$},
author={Pablo Shmerkin and Hong Wang},
year={2021}
}
• Published 16 December 2021
• Mathematics
We establish the dimension version of Falconer’s distance set conjecture for sets of equal Hausdorff and packing dimension (in particular, for Ahlfors-regular sets) in all ambient dimensions. In dimensions d = 2 or 3, we obtain the first explicit estimates for the dimensions of distance sets of general Borel sets of dimension d/2; for example, we show that the set of distances spanned by a planar Borel set of Hausdorff dimension 1 has Hausdorff dimension at least ( √ 5 − 1)/2 ≈ 0.618. In higher…
4 Citations
• Mathematics
• 2022
A BSTRACT . We provide several new answers on the question: how do radial projections distort the dimension of planar sets? Let X, Y Ă R 2 be non-empty Borel sets. If X is not contained on any line,
• Mathematics
• 2021
Let 0 ď s ď 1 and 0 ď t ď 2. An ps, tq-Furstenberg set is a setK Ă R with the following property: there exists a line set L of Hausdorff dimension dimH L ě t such that dimHpK X q ě s for all  P L.
• Mathematics
• 2022
A BSTRACT . We prove two new exceptional set estimates for radial projections in the plane. If K Ă R 2 is a Borel set with dim H K ą 1 , then dim H t x P R 2 z K : dim H π x p K q ď σ u ď max t 1  σ
• Mathematics, Computer Science
• 2022
This paper establishes non-trivial estimates for the additive discretized energy of X c ∈ C, and proves new explicit upper bounds on the quantity dim H, which leads to considerably shorter proofs over the previous works due to Bourgain and Orponen.

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Let 0 ď s ď 1 and 0 ď t ď 2. An ps, tq-Furstenberg set is a setK Ă R with the following property: there exists a line set L of Hausdorff dimension dimH L ě t such that dimHpK X q ě s for all ` P L.
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