Metric number theory of Fourier coefficients of modular forms

TL;DR

This paper explores how well real numbers can be approximated using Fourier coefficients of modular forms, leveraging the Sato-Tate Conjecture and metric number theory to advance understanding in this area.

Contribution

It applies metric number theory to the approximation of real numbers by Fourier coefficients of newforms, building on recent proof of the Sato-Tate Conjecture.

Findings

Established new bounds for approximation quality

Extended metric number theory methods to modular forms

Connected Fourier coefficients with classical approximation problems

Abstract

We discuss the approximation of real numbers by Fourier coefficients of newforms, following recent work of Alkan, Ford and Zaharescu. The main tools used here, besides the (now proved) Sato-Tate Conjecture, come from metric number theory.