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

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