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In this paper, we study the set where 
$$\begin{eqnarray}A=\{p(n)+2^{n}d~\text{mod}~1:n\geq 1\}\subset [0,1],\end{eqnarray}$$
$p$ is a polynomial with at least one irrational coefficient on non-constant terms, 
$d$ is any real number and, for 
$a\in [0,\infty )$, 
$a~\text{mod}~1$ is the fractional part of 
$a$. With the help of a method recently introduced by Wu, we show that the closure of 
$A$ must have full Hausdorff dimension.
