Let $q\geq 2$ be an integer and $\alpha \in R$ such that $(q-1)\alpha \in R \setminus Z.$ For every positve integer $n$ there exists a unique sequence $(a_j(n)_{j\geq 0},$ $a_j(n) \in \{0,1,...,q-1\}$ such that $$n=\sum_{k=0}^{\infty} a_k(n)q^{k}.$$ Define the function sum of digits in the base $q$ by $$S_q(n):=\sum_{k=0}^{\infty} a_k(n).$$ Gelfond alluded the problem of giving an estimate for the number of values of a polynomial $P$ ($P$ takes only integer values on the set $N$) satisfying the condition $S(P(n))\equiv r \mod m,$ where $m\geq 2$ and $r\in Z.$ Morgenbesser answered the question of Gelfond in case of $P(n)=n$ and proved that $$\left|\sum_{n=1}^{N} \exp(2i\pi \alpha S(n))\right|=O(N^{\lambda}),$$ where $\lambda <1.$ Mauduit and Rivat answered the question for $P(n)=n^2$ and showed that $$\left|\sum_{n\leq x} \exp(2i\pi \alpha S(n^2))\right|=O((\log{x})^{(1/2)\omega(q)+4}) x^{1-\sigma_q(\alpha)},$$ where $\sigma_q(\alpha)>0.$ Now, suppose that $P(n)=n^{a},$ $a\geq 3.$ Is there a formula analoguous to the above formulas that can bound the following sum $$\sum_{n\leq x} \exp(2i\pi \alpha S(n^a)) $$ and therefore that answers the Gelfond's question in the case of $P(n)=n^a,$ with $a$ is an integer greater than $2.$? Many thanks.