Finite configurations in sparse sets

DOI10.1007/S11854-016-0010-3zbMATH Open1375.28008arXiv1307.1174OpenAlexW1827651941MaRDI QIDQ279766

Izabella Łaba, Malabika Pramanik, Vincent M. K. Chan

Publication date: 29 April 2016

Published in: Journal d'Analyse Mathématique (Search for Journal in Brave)

Abstract: Let $Esubseteq{R}^n$ be a closed set of Hausdorff dimension $alpha$. For $mgeqn$, let $B_1,ldots,B_k$ be $nimes(m-n)$ matrices. We prove that if the system of matrices $B_j$ is non-degenerate in a suitable sense, $alpha$ is sufficiently close to $n$, and if $E$ supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then for a range of $m$ depending on $n$ and $k$, the set $E$ contains a translate of a non-trivial $k$-point configuration $B_{1}y,ldots,B_{k}y$. As a consequence, we are able to establish existence of certain geometric configurations in Salem sets (such as parallelograms in $R^n$ and isosceles right triangles in $R^2$). This can be viewed as a multidimensional analogue of an earlier result of Laba and Pramanik on 3-term arithmetic progressions in subsets of $R$.

Full work available at URL: https://arxiv.org/abs/1307.1174

zbMATH Keywords

Salem setssparse setsFourier decay conditiongeometric configurations

Mathematics Subject Classification ID

Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type (42B10) Hausdorff and packing measures (28A78)

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