Finite configurations in sparse sets

Abstract

Let $E \subseteq R^n$ be a closed set of Hausdorff dimension $\alpha$. For $m \geq n$, let $\{B_1,\ldots,B_k\}$ 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 $\{B_1y,\ldots,B_ky\}$. 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 $R$.