Distribution of Powers Modulo 1 and Related Topics

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 $n\ge n_0$, then the ${\alpha }_i$'s are algebraic integers. Finally we present several open problem and topics for further research.

arXiv comment18 pages

Cited by in corpus (1)

• On the Uniformity of $(3/2{)}^n$ Modulo 1