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

Abstract

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 $\kappa(z;n)=\lfloor n^z\rfloor$.