Finite configurations in sparse sets

Abstract

Let $E\subseteq R^n$ be a closed set of Hausdorff dimension $\alpha$. For $m\ge n$, let $\{B_1,\dots ,B_k\}$ be $n\times (m-n)$ 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 $\{B_{1}y,\dots ,B_{k}y\}$. 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 $R$.

46 pages

References in corpus (2)

• Arithmetic progressions in sets of fractional dimension
• The true complexity of a system of linear equations

Cited by in corpus (3)

• Rigidity, graphs and Hausdorff dimension
• Equilateral triangles in subsets of ${\mathbb{R}}^d$ of large Hausdorff dimension
• On necklaces inside thin subsets of ${\mathbb{R}}^d$