Algebraic Numbers, Hyperbolicity, and Density Modulo One

TL;DR

This paper proves the density modulo one of certain exponential sums involving algebraic numbers, using dynamics of higher-rank actions on compact abelian groups, advancing understanding of algebraic and dynamical systems.

Contribution

It establishes the density of specific exponential sets modulo one for multiplicatively independent algebraic numbers, employing novel dynamical systems techniques.

Findings

Sets of the form sum of products of algebraic numbers are dense modulo one.

The proof utilizes dynamics of higher-rank actions on compact abelian groups.

Advances the connection between algebraic number theory and dynamical systems.

Abstract

We prove the density of the sets of the form ${{\lambda}_1^m {\mu}_1^n {\xi}_1 +...+{\lambda}_k^m {\mu}_k^n {\xi}_k : m,n \in \mathbb N}$ modulo one, where $\lambda_i$ and $\mu_i$ are multiplicatively independent algebraic numbers satisfying some additional assumptions. The proof is based on analysing dynamics of higher-rank actions on compact abelean groups.