Bennett, Michael A. An ideal Waring problem with restricted summands. (English) Zbl 0793.11026 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.) Cited in 1 ReviewCited in 2 Documents 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 PDF BibTeX XML Cite \textit{M. A. Bennett}, Acta Arith. 66, No. 2, 125--132 (1994; Zbl 0793.11026) Full Text: DOI EuDML OpenURL