Rational approximation with chosen numerators

TL;DR

This paper investigates rational approximation of real numbers using prime denominators with fixed numerators, exploring probabilistic and deterministic methods, and connecting to twisted diophantine approximation and trace functions.

Contribution

It introduces new results on rational approximation with prime denominators and fixed numerators, linking to twisted diophantine approximation and dynamical sequences.

Findings

Established basic probabilistic and deterministic results

Connected rational approximation to twisted diophantine approximation

Presented an application related to trace functions and dynamical sequences

Abstract

We consider the problem of approaching real numbers with rational numbers with prime denominator and with a single numerator allowed for each denominator. We obtain basic results, both probabilistic and deterministic, draw connections to twisted diophantine approximation, and present a simple application, related to possible correlations between trace functions and dynamical sequences.