On Waring’s problem for cubes. (English) Zbl 0574.10046

Vaughan, R. C.

J. Reine Angew. Math. 365, 122-170 (1986).

The author obtains the following results:
(a) The asymptotic formula for the number of representations of integers N as sums of 8 positive cubes. Previously (using Hua’s inequality), the asymptotic formula was established for 9 or more cubes. In the author’s result, the error term differs from the main term by a power of log N.
(b) A lower bound for the number of representations of N as the sum of 7 cubes. This provides a proof of the 7-cubes theorem by the use of the Hardy-Littlewood method.
(c) The number of integers \(\leq N\) that are sums of 3 cubes is \(\gg N^{(19/21)-\epsilon}.\) There are also other related results. Of these, (c) is an improvement of the author’s result [Bull. Lond. Math. Soc. 17, 17-20 (1985; Zbl 0562.10022)], the method for which was derived from the reviewer’s papers [Acta Arith. (to appear; see the following review)] (this being the iterative use of the Hardy-Littlewood method).
The main additional idea required in the present paper lies in a suitable adaptation of an identity of Davenport used in his work on sums of three cubes. The identity is, with \(x_ 2=x_ 1+hm^ 3\), \(4(x^ 3_ 2-x^ 3_ 1)=hm^ 3\{3(2x_ 1+hm^ 3)^ 2+h^ 2m^ 6\}.\) Some ideas of C. Hooley (used in his work on cubes) also are used. The author uses the Hardy-Littlewood method in different ways (especially for estimating the number of solutions of certain equations). The numerous notations have resulted in some misprints.

Reviewer: K.Thanigasalam

MSC:

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

Keywords:

sums of cubes; sums of eight cubes; asymptotic formula; number of representations of integers; lower bound; 7-cubes theorem; Hardy- Littlewood method; identity of Davenport; sums of three cubes

Citations:

Zbl 0574.10047; Zbl 0564.10049; Zbl 0562.10022