On the topology of integer polynomials with bounded coefficients

Let $q>1$ be a real number and $m\geq 1$ an integer. Let $Y$ denote the set of number $f(q)$ where  $f$ runs over  the integer polynomials with height not exceeding $m$. In this talk, we consider an old question when  $Y$ is dense in the real line. This question is closely related to  the studies of Bernoulli convolutions,  beta-expansions  and iterated functions systems. We prove the following conjecture of Erd\H{o}s et al.:  $Y$ is dense if and only if  $q$ is less than $m+1$ and is non-Pisot.