Rational approximation with chosen numerators

TL;DR

This paper investigates approximating real numbers using rationals with prime denominators and fixed numerators, exploring applications to correlations between trace functions and dynamical sequences.

Contribution

It introduces a novel approach to rational approximation with restricted numerators and denominators, and connects this to trace function and dynamical sequence correlations.

Findings

Developed a method for rational approximation with prime denominators and fixed numerators.

Applied the approximation method to analyze correlations between trace functions and dynamical sequences.

Provided insights into the interplay between number theory and dynamical systems.

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 then present a simple application, related to possible correlations between trace functions and dynamical sequences.