## Slim exceptional sets for sums of cubes.(English)Zbl 1007.11058

Let $${\mathcal N}_s$$ be the set of natural numbers $$n$$ such that for every natural number $$q$$, the congruence $$n\equiv x_1^3+\cdots+x_s^3$$ (mod $$q$$) has a solution with $$(x_1x_2\ldots x_s,q)=1$$. (In the paper, the sets $${\mathcal N}_s$$ are defined explicitly by concrete congruence conditions on $$n$$ for $$s\geq 5$$.) Then define $$E_s(X)$$ to be the number of members in $${\mathcal N}_s$$ not exceeding $$X$$ that cannot be written as the sum of $$s$$ cubes of prime numbers. As the first theorem of this paper, it is established for $$5\leq s\leq 8$$ that one has $$E_s(X)\ll X^{\alpha_s+\varepsilon}$$ for any fixed positive $$\varepsilon$$, with $$\alpha_5=35/36$$, $$\alpha_6=17/18$$, $$\alpha_7=23/36$$ and $$\alpha_8=11/36$$. It should be mentioned here that we already know that $$E_s(X)\ll 1$$ for $$s\geq 9$$, and that recent work of Xiumen Ren could show the corresponding bounds with $$\alpha_s=1-(s-4)/153$$ for $$5\leq s\leq 8$$. The substantial improvement on the bound of $$E_5(X)$$ is concerned with a way of pruning the major arcs in application of the Hardy-Littlewood method. Meanwhile, the above noteworthy bounds of $$E_s(X)$$ for $$6\leq s\leq 8$$ are the results of the novel ideas of the author appearing in his previous paper [Slim exceptional sets for sums of four squares, Proc. Lond. Math. Soc. (3) 85, 1-21 (2002; Zbl 1039.11066)]. Indeed, the main purpose of this paper is to apply the latter ideas to various additive problems concerning sums of cubes.
Besides the above problem, the paper treats exceptional set problems associated with sums of cubes of smooth numbers (which means, natural numbers having only ‘small’ prime factors), and with the conjectured asymptotic formula for the number of representations of natural numbers as the sum of seven cubes. Moreover, as for the density of numbers that cannot be written as the sum of six cubes, a conditional estimate is derived from a hypothesis about a certain representation problem involving four cubes.

### MSC:

 11P05 Waring’s problem and variants 11P32 Goldbach-type theorems; other additive questions involving primes 11P55 Applications of the Hardy-Littlewood method

### Keywords:

sums of cubes; Waring’s problem; Waring-Goldbach problem

Zbl 1039.11066
Full Text: