Distribution of Powers Modulo 1 and Related Topics

2013/12/20 by Lerma, Miguel A.
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1312.5996

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 θ_n, it is possible to get an α (given λ≠ 0), or a λ (given α> 1) such that λαⁿ is close to θ_n modulo 1. We also prove that in a number field, if a combination of powers λ₁ α₁ⁿ + ⋯ + λ_m α_mⁿ has bounded v-adic absolute value (where v is any non-Archimedian place) for n ≥ n₀, then the α_i's are algebraic integers. Finally we present several open problem and topics for further research.

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