On the topology of sums in powers of an algebraic number

Abstract

Let $1<q<2$ and \[ Λ(q)={\sum_{k=0}^n a_kq^k\mid a_k\in\{-1,0,1\}, n\ge1}. \] It is well known that if $q$ is not a root of a polynomial with coefficients $0,\pm 1$, then $\Lambda (q)$ is dense in $\mathbb{R}$. We give several sufficient conditions for the denseness of $\Lambda (q)$ when $q$ is a root of such a polynomial. In particular, we prove that if $q$ is not a Perron number or it has a conjugate $\alpha$ such that $q|\alpha |<1$, then $\Lambda (q)$ is dense in $\mathbb{R}$.

arXiv comment10 pages, no figures

Cited by in corpus (1)

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