Vol. 9, No. 3, 2016

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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
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 d, d 2, is greater than 1 2(d + 1) then, for each k +, there exists a nonempty interval I such that, given any sequence {t1,t2,,tk : tj I}, there exists a sequence of distinct points {xj}j=1k+1 such that xj E and |xi+1 xi| = tj for 1 i 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