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arXiv · 0909.3324

On the topology of sums in powers of an algebraic number

Nikita Sidorov·Boris Solomyak
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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 $0,\pm1$, then $Λ(q)$ is dense in $\mathbb{R}$. We give several sufficient conditions for the denseness of $Λ(q)$ 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 $α$ such that $q|α|<1$, then $Λ(q)$ is dense in $\mathbb{R}$.

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Nikita Sidorov, Boris Solomyak. 2011-07-18. On the topology of sums in powers of an algebraic number. https://doi.org/10.4064/aa149-4-3

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