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 , , is greater than then, for each , there exists a nonempty interval such that, given any sequence , there exists a sequence of distinct points such that and for . In other words, contains vertices of a chain of arbitrary length with prescribed gaps.
Citation
Michael Bennett. Alexander Iosevich. Krystal Taylor. "Finite chains inside thin subsets of $\mathbb{R}^d$." Anal. PDE 9 (3) 597 - 614, 2016. https://doi.org/10.2140/apde.2016.9.597
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