The sum of digits of polynomial values in arithmetic progressions

Let q, m≥ 2 be integers with $(m,q-1)=1$. Denote by sq(n) the sum of digits of n in the q-ary digital expansion. Further let p(x)∈ Z[x] be a polynomial of degree h≥ 3 with p(N)⊂ N. We show that there exist $C=C(q,m,p)>0$ and N₀=N₀(q,m,p)≥ 1, such that for all g and all N≥ N₀, #\0≤ n< N: sq(p(n))≡ g m\≥ C N4/(3h+1). This is an improvement over the general lower bound given by Dartyge and Tenenbaum (2006), which is C N2/h!.