Distribution of Powers Modulo 1 and Related Topics

TL;DR

This paper reviews results on the distribution of powers modulo 1, including approximation properties, algebraic integer conditions, and open problems in the field.

Contribution

It provides new proofs of approximation results, characterizes algebraic integers via boundedness in number fields, and discusses open problems in distribution of powers modulo 1.

Findings

Existence of $ heta_n$ approximations for given sequences

Characterization of algebraic integers through bounded $v$-adic values

Presentation of open problems and future research directions

Abstract

This is a review of several results related to distribution of powers and combination of powers modulo 1. We include a proof that given a sequence of real numbers $\theta_n$, it is possible to get an $\alpha$ (given $\lambda \ne 0$), or a $\lambda$ (given $\alpha > 1$) such that $\lambda \alpha^n$ is close to $\theta_n$ modulo 1. We also prove that in a number field, if a combination of powers $\lambda_1 \alpha_1^n + \cdots + \lambda_m \alpha_m^n$ has bounded $v$-adic absolute value (where $v$ is any non-Archimedian place) for $n \geq n_0$, then the $\alpha_i$'s are algebraic integers. Finally we present several open problem and topics for further research.