The sum of digits of polynomial values in arithmetic progressions

Thomas Stoll 1, *
Abstract : 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!).
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Thomas Stoll. The sum of digits of polynomial values in arithmetic progressions. Functiones et Approximatio Commentarii Mathematici, Poznań : Wydawnictwo Naukowe Uniwersytet im. Adama Mickiewicza, 2012, 47 (2), pp.233-239. 〈http://projecteuclid.org/euclid.facm/1356012917〉. 〈10.7169/facm/2012.47.2.7〉. 〈hal-01278713〉

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