Rendiconti del Seminario Matematico della Università di Padova


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Volume 136, 2016, pp. 205–223
DOI: 10.4171/RSMUP/136-14

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

Hajme Kaneko[1]

(1) Institute of Mathematics, University of Tsukuba, 1-1-1, Tennodai, 305-8571, Tsukuba, Ibaraki, Japan

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

Keywords: Beta expansions, nonzero digits, Pisot numbers, Salem numbers

Kaneko Hajme: On the number of nonzero digits in the beta-expansions of algebraic numbers. Rend. Sem. Mat. Univ. Padova 136 (2016), 205-223. doi: 10.4171/RSMUP/136-14