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Further improvements in Waring’s problem. IV: Higher powers. (English) Zbl 0972.11092

As usual, \(G(k)\) is to denote the least \(s\) such that every sufficiently large integer is expressible as a sum of at most \(s\) \(k\)th powers of positive integers. This paper describes new bounds for \(G(k)\) for \(9 \leq k \leq 20\), ranging from \(G(9) \leq 50\) to \(G(20) \leq 142\). The value for \(G(9)\) improves (by 1) a result from the previous papers I, II, III of this series [Acta Math. 174, 147-240 (1995; Zbl 0849.11075); Duke Math. J. 76, 683-710 (1994; Zbl 0849.11076); Philos. Trans. R. Soc. Lond., Ser. A 345, 385-396 (1993; Zbl 0849.11077), (these papers appeared in the opposite order to that indicated by their titles]. The value for \(G(20)\) improves (again by 1) the result by Zaizhao Meng [J. China Univ. Sci. Technol. 27, No. 1, 1-5 (1997; Zbl 0906.11049)], this being the only instance known to the authors of a result published subsequent to paper I and to Wooley’s paper on “smooth” Weyl sums [J. Lond. Math. Soc., II. Ser. 51, 1-13 (1995; Zbl 0833.11041)] that improved the results described in these papers.
Following R. C. Vaughan’s paper [Acta Math. 162, 1-71 (1989; Zbl 0665.10033)], the authors use upper bounds for the number of solutions of an auxiliary Diophantine equation \(\sum_{1 \leq i \leq s}x_i^k = \sum_{1 \leq i \leq s}y_i^k\), in which the numbers \(x_i\), \(y_i\) are to be “smooth” (or “friable”, as it is sometimes termed), so that all of its prime factors are less than a specified bound. Further, they appeal to estimates from Wooley’s paper on smooth Weyl sums. They also use technical refinements of ideas from the earlier papers in the current series. These ideas lead to substantial calculations before the results can be attained.
Following a paper of T. D. Wooley [Acta Arith. 65, 163-179 (1993; Zbl 0789.11037)] the work on the solutions of the auxiliary equation also lead to improved estimates for \(\min_{1 \leq n \leq N}\{\alpha n^k\}\), where \(\{\cdot\}\) denotes the fractional part. When \(k \leq 7\), however, the results of D. R. Heath-Brown [Mathematika 35, 28-37 (1988; Zbl 0629.10029)] remain the sharper ones.

MSC:

11P05 Waring’s problem and variants
11J71 Distribution modulo one
11P55 Applications of the Hardy-Littlewood method
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