Finite chains inside thin subsets of ℝd

Bennett, Michael; Iosevich, Alexander; Taylor, Krystal

doi:10.2140/apde.2016.9.597

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Finite chains inside thin subsets of $\mathbb{R}^d$

Michael Bennett, Alexander Iosevich and Krystal Taylor

Vol. 9 (2016), No. 3, 597–614

DOI: 10.2140/apde.2016.9.597

Abstract

In a recent paper, Chan, Łaba, and Pramanik investigated geometric configurations inside thin subsets of Euclidean space possessing measures with Fourier decay properties. In this paper we ask which configurations can be found inside thin sets of a given Hausdorff dimension without any additional assumptions on the structure. We prove that if the Hausdorff dimension of $E \subset ℝ^{d}$ , $d \geq 2$ , is greater than $\frac{1}{2} (d + 1)$ then, for each $k \in ℤ^{+}$ , there exists a nonempty interval $I$ such that, given any sequence ${t_{1}, t_{2}, \dots, t_{k} : t_{j} \in I}$ , there exists a sequence of distinct points ${x^{j}}_{j = 1}^{k + 1}$ such that $x^{j} \in E$ and $| x^{i + 1} - x^{i} | = t_{j}$ for $1 \leq i \leq k$ . In other words, $E$ contains vertices of a chain of arbitrary length with prescribed gaps.

Keywords

classical analysis and ODEs, combinatorics, metric geometry, chains, geometric measure theory, geometric configurations, Hausdorff dimension, Falconer distance problem

Mathematical Subject Classification 2010

Primary: 28A75, 42B10

Secondary: 53C10

Milestones

Received: 10 September 2014

Revised: 23 April 2015

Accepted: 11 October 2015

Published: 17 June 2016

Authors

Michael Bennett
Department of Mathematics University of Rochester 915 Ray P. Hylan Building 145 Dunrovin Lane Rochester, NY 14628 United States

Alexander Iosevich
Department of Mathematics University of Rochester 915 Ray P. Hylan Building 145 Dunrovin Lane Rochester, NY 14628 United States

Krystal Taylor
Department of Mathematics The Ohio State University 231 W. 18th Ave., MW 706 Columbus, OH 43210 United States

Vol. 9, No. 3, 2016