Sets of large dimension not containing polynomial configurations

Abstract

The main result of this paper is the following. Given countably many multivariate polynomials with rational coefficients and maximum degree $d$, we construct a compact set $E\subset {\mathbb{R}}^n$ of Hausdorff dimension $n/d$ which does not contain finite point configurations corresponding to the zero sets of the given polynomials. Given a set $E\subset {\mathbb{R}}^n$, we study the angles determined by three points of $E$. The main result implies the existence of a compact set in ${\mathbb{R}}^n$ of Hausdorff dimension $n/2$ which does not contain the angle $\pi /2$. (This is known to be sharp if $n$ is even.) We show that there is a compact set of Hausdorff dimension $n/8$ which does not contain an angle in any given countable set. We also construct a compact set $E\subset {\mathbb{R}}^n$ of Hausdorff dimension $n/6$ for which the set of angles determined by $E$ is Lebesgue null. In the other direction, we present a result that every set of sufficiently large dimension contains an angle $\epsilon$ close to any given angle.