Journal of the European Mathematical Society


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Volume 18, Issue 1, 2016, pp. 181–193
DOI: 10.4171/JEMS/587

Published online: 2015-12-16

On the topology of polynomials with bounded integer coefficients

De-Jun Feng[1]

(1) The Chinese University of Hong Kong, Shatin, Hong Kong, China

For a real number $q>1$ and a positive integer $m$, let $$ Y_m(q):=\left\{\sum_{i=0}^n\epsilon_i q^i:\; \epsilon_i\in \{0, \pm 1,\ldots, \pm m\},\; n=0, 1,\ldots \right\}. $$ In this paper, we show that $Y_m(q)$ is dense in $\mathbb R$ if and only if $q < m + 1$ and $q$ is not a Pisot number. This completes several previous results and answers an open question raised by Erdős, Joó and Komornik [8].

Keywords: Pisot numbers, iterated function systems

Feng De-Jun: On the topology of polynomials with bounded integer coefficients. J. Eur. Math. Soc. 18 (2016), 181-193. doi: 10.4171/JEMS/587