JournalsrsmupVol. 136pp. 205–223

On the number of nonzero digits in the beta-expansions of algebraic numbers

  • Hajme Kaneko

    University of Tsukuba, Tsukuba, Ibaraki, Japan
On the number of nonzero digits in the beta-expansions of algebraic numbers cover
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Abstract

Many mathematicians have investigated the base-bb expansions for integral base-b2b \geq 2, and more general β\beta-expansions for a real number β>1\beta > 1. However, little is known on the β\beta-expansions of algebraic numbers. The main purpose of this paper is to give new lower bounds for the numbers of nonzero digits in the β\beta-expansions of algebraic numbers under the assumption that β\beta is a Pisot or Salem number. As a consequence of our main results, we study the arithmetical properties of power series n=1βκ(z;n)\sum_{n=1}^{\infty} \beta^{-\kappa(z;n)}, where z>1z > 1 is a real number and κ(z;n)=nz\kappa(z;n)=\lfloor n^z\rfloor.

Cite this article

Hajme Kaneko, On the number of nonzero digits in the beta-expansions of algebraic numbers. Rend. Sem. Mat. Univ. Padova 136 (2016), pp. 205–223

DOI 10.4171/RSMUP/136-14