On Waring's problem for two cubes and two small cubes (Q2919687)

It is conjectured that every sufficiently large positive integer can be represented as the sum of four positive cubes. This is known to be true for almost all positive integers of size at most \(N\) except possibly a set of size \(N^{37/42-\tau}\) for some small \(\tau>0\), see for example [\textit{K. Kawada} and \textit{T. D. Wooley}, J. Lond. Math. Soc., II. Ser. 82, No. 2, 437--458 (2010; Zbl 1279.11097)].NEWLINENEWLINEIn this paper, the author studies the representation function \(\rho_\theta(n,P)\), that counts the number of solutions to the equation \(n=x^3+y^3+z_1^3+z_2^3\), with \(x\sim P\), \(y\sim P^{5/6}\) and \(z_1,z_2\sim \frac{1}{2}P^{3\theta}\) for some \(\theta >0\). The main result of the paper states that for every \(7/36<\theta\leq 2/9\) there is some \(\delta>0\) such that the mean square estimate NEWLINE\[NEWLINE \sum_{P^3\leq n\leq 2P^3} (\rho_\theta(n,P)-\frac{1}{3}\mathfrak{S}(n) QR^2n^{-2/3})^2\ll P^{12\theta+2/3-\delta} NEWLINE\]NEWLINE holds; here \(\mathfrak{S}(n)\) is the familiar singular seriesNEWLINEassociated with sums of four cubes.NEWLINENEWLINE Hence one obtains a lower bound for \(\rho_\theta(n,P)\) of the expected order of magnitude for almost all positive integers \(n\). As a corollary of this result one deduces in the same range of \(\theta\) that almost all positive integers \(n\) may be represented as a sum of four positive cubes, where two of the cubes are restricted in size to \(z_j\leq n^\theta\), \(j=1,2\). This improves on previous work of \textit{S.-L. A. Lee} [Acta Arith. 151, No. 4, 377--399 (2012; Zbl 1315.11088)], who obtained this conclusion only for \(\theta\geq \frac{192}{869}\).NEWLINENEWLINEAdding the exponents of the size restrictions for \(x,y,z_1,z_2\) in the counting function \(\rho_\theta(n,P)\), the main result of the paper can be interpreted as the statement that almost all natural numbers can be represented as the sum of \(3+\varepsilon\) cubes. This is best possible in a sense that \(3\) positive cubes are not sufficient due to a local obstruction modulo \(9\).NEWLINENEWLINEThe proof of the main result proceeds via an application of the Hardy-Littlewood circle method. It uses the technique of diminishing ranges in contrast to the \(p\)-adic iterative method implicitly used in Lee's work.