The sum of digits of polynomial values in arithmetic progressions

Overview & Synopsis

Let q, m ≥ 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) ∈ 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 N_0 = N_0(q, m, p) ≥ 1, such that for all g ∈ Z and all N ≥ N 0 , #{0 ≤ n < N : s_q (p(n)) ≡ g mod m} ≥ CN^(4/(3h+1)). This is an improvement over the general lower bound given by Dartyge and Tenenbaum (2006), which is CN^(2/h!).