Distribution of Powers Modulo 1 and Related Topics
Miguel A. Lerma

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 , it is possible to get an (given ), or a (given ) such that is close to modulo 1. We also prove that in a number field, if a combination of powers has bounded -adic absolute value (where is any non-Archimedian place) for , then the 's are algebraic integers. Finally we present several open problem and topics for further research.
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Taxonomy
Topicsadvanced mathematical theories
