On the topology of polynomials with bounded integer coefficients

De-Jun Feng

doi:10.4171/jems/587

JournalsjemsVol. 18, No. 1pp. 181–193

On the topology of polynomials with bounded integer coefficients

De-Jun Feng
The Chinese University of Hong Kong, Shatin, Hong Kong, China

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Abstract

For a real number $q > 1$ and a positive integer $m$ , let

Y_{m} (q) := {i = 0 \sum n ϵ_{i} q^{i} : ϵ_{i} \in {0, \pm 1, \dots, \pm m}, n = 0, 1, \dots} .

In this paper, we show that $Y_{m} (q)$ is dense in $R$ 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].

Cite this article

De-Jun Feng, On the topology of polynomials with bounded integer coefficients. J. Eur. Math. Soc. 18 (2016), no. 1, pp. 181–193

DOI 10.4171/JEMS/587