Analysis & PDE

  • Anal. PDE
  • Volume 9, Number 3 (2016), 597-614.

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

Michael Bennett, Alexander Iosevich, and Krystal Taylor

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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.

Article information

Anal. PDE, Volume 9, Number 3 (2016), 597-614.

Received: 10 September 2014
Revised: 23 April 2015
Accepted: 11 October 2015
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 28A75: Length, area, volume, other geometric measure theory [See also 26B15, 49Q15] 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
Secondary: 53C10: $G$-structures

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


Bennett, Michael; Iosevich, Alexander; Taylor, Krystal. Finite chains inside thin subsets of $\mathbb{R}^d$. Anal. PDE 9 (2016), no. 3, 597--614. doi:10.2140/apde.2016.9.597.

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