Metric results on sumsets and Cartesian products of classes of Diophantine sets

Abstract

Erdős proved that any real number can be written as a sum, and a product, of two Liouville numbers. Motivated by these results, we study sumsets of classes of real numbers with prescribed (or bounded) irrationality exponents. We show that such sumsets turn out to be large in general, indeed almost every real number with respect to Lebesgue measure can be written as the sum of two numbers with sufficiently large prescribed irrationality exponents. In fact the Hausdorff dimension of the complement is small, and the result remains true if we impose considerably refined conditions on the orders of rational approximation (``exact approximation'' with respect to an approximation function). As an application, we show that in many cases the Hausdorff dimension of Cartesian products of sets with prescribed irrationality exponent exceeds the expected dimension, that is the sum of the single Hausdorff dimensions. We also address their packing dimensions. Similar results hold when restricting to classical missing digit Cantor sets, relative to its natural Cantor measure. In particular, we prove that the subset of numbers with prescribed large irrationality exponent has full packing dimension, i.e. the same packing dimension as the entire Cantor set. This complements some results on the Hausdorff dimension of these sets, which is an extensively studied topic in Diophantine approximation. Our proofs are based on ideas of Erdős, but vastly extend them.