On the topology of polynomials with bounded integer coefficients

For a real number $q>1$ and a positive integer $m$, let $Y_m(q):={\sum_{i=0}^nε_i q^i:\; ε_i\in \{0, \pm 1,..., \pm m\}, n=0, 1,...}.$ In this paper, we show that $Y_m(q)$ is dense in ${\Bbb R}$ 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 Erdös, Joó and Komornik.

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