- 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.