Oliver Pfeiffer: Waring's problem with sum of digits congruence restrictions

This is a joint work with Jörg Thuswaldner.

Waring's Problem, conjecturing that every integer $ N$ can be respresented as sum $ N=n_1^d+\ldots+n_s^d$ of a sufficiently large number of powers of other integers, is investigated subject to so-called digital restrictions. That is, the indeterminates $ n_1$, ..., $ n_s$ all obey a congruence of the type $ s_q(n)\equiv a(m)$, where $ s_q$ denotes the $ q$-adic sum of digits function. Given $ N$, $ s$, $ d$, $ a$, $ m$ and $ q$, we provide a Hardy-Littlewood like asymptotic formula for the number of such representations of $ N$, from which the fact that the set $ \{n\geq1:s_q(n)\equiv a(m)\}$ forms an asymptotic basis can be easily derived. This is done for a single digital restriction (Thuswaldner and Tichy [3]) as well as for systems of sum of digits congruences (Pfeiffer and Thuswaldner [2]) using methods introduced by Kim [1].


Dong-Hyun Kim.
On the joint distribution of $ q$-additive functions in residue classes.
J. Number Theory, 74(2):307-336, 1999.

Oliver Pfeiffer and Jörg M. Thuswaldner.
Waring's problem restricted by a system of sum of digits congruences.

Jörg M. Thuswaldner and Robert F. Tichy.
Waring's problem with digital restrictions.
(to appear).

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