On the topology of polynomials with bounded integer coefficients

De-Jun Feng

On the topology of polynomials with bounded integer coefficients

Number Theory 2015-02-03 v3 Classical Analysis and ODEs

Authors: De-Jun Feng

Abstract

For a real number $q>1$ and a positive integer $m$ , let $Y_m(q):={\sum_{i=0}^n\epsilon_i q^i:\; \epsilon_i\in \{0, \pm 1,..., \pm m\}, n=0, 1,...}.$ 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 Erd\"{o}s, Jo\'{o} and Komornik.

Keywords

additive number theory polynomial theory positive polynomials

Cite

@article{arxiv.1109.1407,
  title  = {On the topology of polynomials with bounded integer coefficients},
  author = {De-Jun Feng},
  journal= {arXiv preprint arXiv:1109.1407},
  year   = {2015}
}

Comments

15 pages, to appear in J. Eur. Math. Soc

On the topology of polynomials with bounded integer coefficients

Abstract

Keywords

Cite

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