An additive problem with fractional powers
For \(u\in\mathbb R\), denote by \([u]\) and \(\|u\|\) the largest integer not exceeding \(u\) and the distance from \(u\) to the nearest integer, respectively. \textit{J.-M. Deshouillers} [Acta Arith. 25, 393--403 (1973--1974; Zbl 0278.10020)] proved that for all \(\alpha\in(1,4/3)\) any sufficiently large natural number \(x\) can be expressed as \(x=[U^\alpha]+ [V^\alpha]\) with nonnegative integers \(U\) and \(V\). \textit{S. A. Gritsenko} [Russ. Acad. Sci., Izv., Math. 41, No. 3, 447--464 (1993); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 56, No. 6, 1198--1216 (1992; Zbl 0810.11057)] extended the range of \(\alpha\) possessing this property to \(\alpha\in(1,55/41)\). \textit{M. Z. Garaev} and \textit{K. L. Kuch} [J. Anal. Appl. 20, 999--1006 (2001; Zbl 1030.11037)] showed that for all \(\alpha\in(1,3/2]\) and \(N\geq 2\) there exist \(\gg N\) natural numbers \(x\leq N\) expressibile as \(x=[U^\alpha]+[V^\alpha]\), where \(U\) and \(V\) are natural numbers. Here the author establishes the following result. Theorem. For all \(\alpha\in(1,3/2)\), any sufficiently large natural number \(x\) can be expressed as \(x=[U^\alpha]+ [V^\alpha]\), where \(U\) and \(V\) are natural numbers.
scientific article; zbMATH DE number 2115607