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 ⊂ 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 ⊂ R n , we study the angles determined by three-point subsets of E . The main result implies the existence of a compact set in R n of Hausdorff dimension n / 2 which does not contain the angle π / 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 ⊂ 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 e close to any given angle. The main result can also be applied to distance sets. As a corollary we obtain a compact set E ⊂ R n ( n ≥ 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 .