On the densities of rational multiples

TL;DR

This paper extends the Erdos-Davenport theorem to rational multiples, showing that sets of rational multiples of any natural number set have a well-defined logarithmic density.

Contribution

It proves an analogous logarithmic density result for sets of rational multiples, expanding the classical theorem to a broader context.

Findings

Sets of rational multiples have a logarithmic density similar to natural number multiples.

The theorem generalizes the Erdos-Davenport result to rational numbers.

Provides a new understanding of the distribution of rational multiples.

Abstract

The Erdos-Davenport theorem on the multiples claims that for any set of natural numbers the set consisting of their multiples possesses the logarithmic density. An analogous statement is proved for the sets of rational multiples.