95-292 Miguel A. Lerma
Distribution of Powers Modulo 1 and Related Topics (50K, AMS-TeX 1.2 beta) Jun 21, 95
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Abstract. This is a review of several results related to distribution of powers and combinations of powers modulo~1. We include a proof that given any sequence of real numbers $\theta_n$, it is possible to get an $\alpha$ (given $\lambda \neq 0$), 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 $\lambda_1\,\alpha_1^n + \dots + \lambda_m\,\alpha_m^n$ has bounded $v$-adic absolute value (where $v$ is any non-Archimedean place) for $n \geq n_0$, then the $\alpha_i$'s are $v$-adic algebraic integers. Finally we present several open problems and topics for further research.

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