Bernoulli decomposition and arithmetical independence between sequences

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.