The sum of digits of polynomial values in arithmetic progressions

Let $q, m\geq 2$ be integers with $(m,q-1)=1$. Denote by $s_q(n)$ the sum of digits of $n$ in the $q$-ary digital expansion. Further let $p(x)\in mathbb{Z}[x]$ be a polynomial of degree $h\geq 3$ with $p(\mathbb{N})\subset \mathbb{N}$. We show that there exist $C=C(q,m,p)>0$ and $N_0=N_0(q,m,p)\geq 1$, such that for all $g\in\mathbb{Z}$ and all $N\geq N_0$, $$#\{0\leq n< N: \quad s_q(p(n))\equiv g \bmod m\}\geq C N^{4/(3h+1)}.$$ This is an improvement over the general lower bound given by Dartyge and Tenenbaum (2006), which is $C N^{2/h!}$.

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