#### 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 $E\subset {ℝ}^{d}$, $d\ge 2$, is greater than $\frac{1}{2}\left(d+1\right)$ then, for each $k\in {ℤ}^{+}$, there exists a nonempty interval $I$ such that, given any sequence $\left\{{t}_{1},{t}_{2},\dots ,{t}_{k}:{t}_{j}\in I\right\}$, there exists a sequence of distinct points ${\left\{{x}^{j}\right\}}_{j=1}^{k+1}$ such that ${x}^{j}\in E$ and $|{x}^{i+1}-{x}^{i}|={t}_{j}$ for $1\le i\le k$. 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