Sets of large dimension not containing polynomial configurations

Máthé, András (2017) Sets of large dimension not containing polynomial configurations. ADVANCES IN MATHEMATICS, 316. pp. 691-709. ISSN 0001-8708

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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⊂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-point subsets 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. © 2017

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