On an open problem of Gelfond

Question

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.

Answer 1

Michael Drmota and Christian Mauduit [EDIT: and Joel Rivat], The sum-of-digits function of polynomial sequences, J. Lond. Math. Soc. (2) 84 (2011), no. 1, 81–102, MR2819691 (2012f:11193) Theorem 1:

Let $d\ge2$ be an integer, $q\ge q_0(d)$ be a sufficiently large prime number, and $P$ a polynomial of degree $d$ with integer coefficients and taking natural numbers to natural numbers, with leading coefficient $a_d$ coprime to $q$. If $(q-1)\alpha$ is real but not an integer, then there exists $\sigma>0$ such that $$\sum_{n<x}e^{2\pi i\alpha s_q(P(n))}\ll x^{1-\sigma}$$ where the implied constant depends on $q$, $d$, and $\alpha$.