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Erd\H{o}s 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 ofclasses of real numbers with prescribed (or bounded) irrationality exponents.We show that such sumsets turn out to be large in general, indeed almost everyreal number with respect to Lebesgue measure can be written as the sum of twonumbers with sufficiently large prescribed irrationality exponents. In fact theHausdorff dimension of the complement is small, and the result remains true ifwe impose considerably refined conditions on the orders of rationalapproximation (``exact approximation'' with respect to an approximationfunction). As an application, we show that in many cases the Hausdorffdimension of Cartesian products of sets with prescribed irrationality exponentexceeds the expected dimension, that is the sum of the single Hausdorffdimensions. We also address their packing dimensions. Similar results hold whenrestricting to classical missing digit Cantor sets, relative to its naturalCantor measure. In particular, we prove that the subset of numbers withprescribed large irrationality exponent has full packing dimension, i.e. thesame packing dimension as the entire Cantor set. This complements some resultson the Hausdorff dimension of these sets, which is an extensively studied topicin Diophantine approximation. Our proofs are based on ideas of Erd\H{o}s, butvastly extend them.

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