On a Notion of Density Related to Restricted Diophantine Approximation

We introduce notion of density for a subset of the natural numbers called the Diophantine density. First, we demonstrate that the Diophantine density is an abstract density in the sense of Di Nasso and Jin. Then we establish upper and lower bounds for the Diophantine density of a set and its complement respectively. We also exhibit a construction which allows us to find a set whose density can be any value between zero and one. Finally, we prove a restricted-denominator analog of Dirichlet’s approximation theorem for sets of non-zero Diophantine density.