An ideal Waring problem with restricted summands. (English) Zbl 0793.11026

Bennett, Michael A.

Acta Arith. 66, No. 2, 125-132 (1994).

It is proved that if \(g_ N(k)\) denotes the order of the set \(\{1^ k, N^ k, (N+1)^ k, \dots\}\) as an additive basis for the positive integers (so that \(g_ N(k)= g(k)\) from Waring’s problem), then we have \[ g_ N(k)= N^ k+ \Biggl[ \biggl( {{N+1}\over N}\biggr)^ k \Biggr]- 2 \] provided \(4\leq N\leq (k+1)^{(k-1)/k} -1\). This result follows from the Hardy-Littlewood-Vinogradov circle method, effective lower bounds upon fractional parts of powers of rationals and an ascent argument due to Dickson.

Reviewer: Michael A. Bennett (Waterloo/Ont.)

MSC:

11P05	Waring’s problem and variants
11P55	Applications of the Hardy-Littlewood method

Keywords:

additive basis; Waring’s problem; Hardy-Littlewood-Vinogradov circle method