Publication date:
20 August 2017
Source:Advances in Mathematics, Volume 316
Author(s): András Máthé
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-point subsets 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 ε close to any given angle. The main result can also be applied to distance sets. As a corollary we obtain a compact set $E\subset {\mathbb{R}}^n$ ($n\ge 2$) of Hausdorff dimension $n/2$ which does not contain rational distances nor collinear points, for which the distance set is Lebesgue null, moreover, every distance and direction is realised only at most once by E.