  In this paper we study the following set\[A=\{p(n)+2^nd \mod 1: n\geq
1\}\subset [0.1],\] where $p$ is a polynomial with at least one irrational
coefficient on non constant terms, $d$ is any real number and for $a\in
[0,\infty)$, $a \mod 1$ is the fractional part of $a$. By a Bernoulli
decomposition method, we show that the closure of $A$ must have full Hausdorff
dimension.
