The exceptional set in Roth's theorem concerning a cube and three cubes of primes (Q2716951)

The author considers the expression of positive integers \(n\) as the sum of a cube and three cubes of primes. Define by \(E(N)\) the cardinality of the exceptional set in this expression. Then a classical result of \textit{K. F. Roth} [Proc. Lond. Math. Soc. (2) 53, 268-279 (1951; Zbl 0043.27303)] states that \(E(N)\ll N\log^{-A}N\) with arbitrary \(A>0\). In this paper Roth's result is improved to \(E(N)\ll N^{169/170}\). NEWLINENEWLINENEWLINEThe improvement is obtained via the circle method and several new ideas. To get a result of this strength, one has to deal with rather big major arcs, to which the Siegel-Walfisz theorem does not apply. In contrast to the previous works, which treat the enlarged major arcs by the Deuring-Heilbronn phenomenon, the author uses the ideas of \textit{T. Zhan} and the reviewer [Sci. China, Ser. A 41, 710-722 (1998; Zbl 0938.11048)] and of \textit{M.-C. Liu, T. Zhan} and the reviewer [Monatsh. Math. 128, 283-313 (1999; Zbl 0940.11047)]. These ideas reveal that in many cases the possible existence of a Siegel zero has no special influence, and hence the Deuring-Heilbronn phenomenon can be avoided. This makes it possible to get better results by using only the mean value estimates, zero-free regions, and zero-density estimates of \(L\)-functions. At the same time, some additional difficulties from the minor arcs are also overcome. NEWLINENEWLINENEWLINEThe result in this paper can also be viewed as an approximation to the conjecture that all sufficiently large integers \(n\) satisfying some necessary congruence conditions are the sum of four cubes of primes. Such a strong result is out of reach at present. However, in a recent paper the author [Chin. Ann. Math., Ser. B 22, No. 2, 233-242 (2001; Zbl 0984.11048)] has proved that a positive proportion of positive integers can be expressed in this way.