The sum of digits of polynomial values in arithmetic progressions

Abstract

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 N 0 = N 0 (q, m, p) ≥ 1, such that for all g ∈ Z and all N ≥ N 0 , #{0 ≤ n < N : sq(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! .

Combinatorial constructions for the Zeckendorf sum of digits of polynomial values

“…Our main result complements the investigation put forward by Dartyge and Tenenbaum [5], and by the author [21], and is as follows. 2 Theorem 1.1.…”

“…We obtain this expansion by using relation (11) a number of (l − 1) times. The following lemma is the key result in the proof of Theorem 1.1 and is an analogue to Lemma 2.1 of [21].…”

On a problem of Chen and Liu concerning the prime power factorization of 𝑛!

“…The constants δ (1) p,m > 0 and δ (2) p,m > 0 can be computed in an effective way; explicit expressions can be found in [12,13]. In contrast, we use an idea of Stoll [16] to obtain general lower bounds for higher-degree powers. The method is constructive.…”

“…Furthermore, the relation [16] says that for all u 1 and Furthermore, the relation [16] says that for all u 1 and…”

On Simultaneous Digital Expansions of Polynomial Values

“…The sum of digits of polynomial values has been at the center of interest in many works. We mention the (still open) conjecture of Gelfond [5] from 1967/68 about the distribution of s q of polynomial values in arithmetic progressions (see also [4,7,10]) and the fundamental work of Bassily and Kátai [1] on central limit theorems satisfied by s q supported on polynomial values resp. polynomial values with prime arguments.…”