Waring's Problem Restricted by a System of Sum of Digits Congruences

Abstract

The aim of the present paper is to generalize earlier work by Thuswaldner and Tichy on Waring's problem with digital restrictions to systems of digital restrictions. Let s_q (n) be the q-adic sum of digits function and let d, s, a_l, m_l , q_l ∈ N. Then for s > d²(log d + log log d + O (1)) there exists N₀ ∈ N such that each integer N ≥N₀ has a representation of the form N = x^d ₁ + ˙ ˙ ˙ + x^d _s where s_ql(xi) ≡ a_l mod m_l (1≤ i ≤ s and 1≤ l ≤ L). The result, together with an asymptotic formula of the number of this representations, will be shown with the help of the circle method together with exponential sum estimates.