Sets of Large Dimension Not Containing Polynomial Configurations

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 ⊂ Rn 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 ⊂ Rn, we study the angles determined by three points of E. The main result implies the existence of a compact set in Rn 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 ⊂ Rn 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 ⊂ Rn (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.