On the topology of polynomials with bounded integer coefficients

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 $\mathbb{R}$ 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 Erdös, Joó and Komornik.

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

References in corpus (1)

• Box-counting fractal strings, zeta functions, and equivalent forms of Minkowski dimension

Cited by in corpus (2)

• When the Weak Separation Condition implies the Generalized Finite Type Condition
• On the cardinality and complexity of the set of codings for self-similar sets with positive Lebesgue measure