On Waring’s problem: two cubes and two minicubes.

*(English)*Zbl 1315.11088From the text: H. Davenport proved in [Acta Math. 71, 123–143 (1939; Zbl 0021.10601)] that almost every natural number can be expressed as a sum of four positive integral cubes. It is now known, courtesy of J. Brüdern [Math. Proc. Camb. Philos. Soc. 109, No. 2, 229–256 (1991; Zbl 0729.11046)] and T. D. Wooley [Mathematika 47, No. 1–2, 53–61 (2000; Zbl 1026.11075)], that when \(N\) is sufficiently large, the number of positive integers at most \(N\) that fail to be written in such a way is slightly smaller than \(N^{37/42}\) Since any integer congruent to \(4\pmod 9\) is never a sum of three cubes, the number of summands here cannot in general be reduced. A heuristic argument shows, however, that one of the four cubes is almost redundant. This motivates the work of Brüdern and Wooley (see [Nagoya Math. J. 200, 59–91 (2010; Zbl 1238.11090)]) on the representation of almost all positive integers as a sum of four cubes, one of which is small (henceforth we call this a minicube). They have shown that, with \(n\) being the natural number to be represented, such a minicube can be as small as \(n^5/36\) without obstructing the existence of representations. This raises the question as to whether we can restrict not only one, but two (or even three) of the cubes in such a representation to be minicubes, and still get an almost all result. The purpose of this paper is to investigate representations of natural numbers by sums of four cubes, two of which are small.

When \(n\) is a positive integer and \(\theta > 0\), write \(r_\theta(n)\) for the number of solutions to the equation \[ n= x_1^3+ x_2^3+ y_1^3+ y_2^3, \]

where \(x_1, x_2, y_1, y_2\) are natural numbers satisfying \(y_1, y_2 \leq n^\theta\).

A formal application of the circle method suggests that when \(1/6 < \theta < 1/3\), we should have \[ r_\theta(n) \sim \frac{\Gamma(4/3)^2}{\Gamma(2/3)} \mathfrak{S}(n) n^{2\theta-1/3}, \] with \(\mathfrak{S}(n)\) being the familiar singular series associated with the representation of positive integers as sums of four cubes.

The following theorem is proved.

Theorem. Whenever \(192/869 \leq \theta \leq 2/9\), we have \(r_\theta(n) \geq 1\) for almost all integers \(n\).

An additional asymptotic formula is established.

When \(n\) is a positive integer and \(\theta > 0\), write \(r_\theta(n)\) for the number of solutions to the equation \[ n= x_1^3+ x_2^3+ y_1^3+ y_2^3, \]

where \(x_1, x_2, y_1, y_2\) are natural numbers satisfying \(y_1, y_2 \leq n^\theta\).

A formal application of the circle method suggests that when \(1/6 < \theta < 1/3\), we should have \[ r_\theta(n) \sim \frac{\Gamma(4/3)^2}{\Gamma(2/3)} \mathfrak{S}(n) n^{2\theta-1/3}, \] with \(\mathfrak{S}(n)\) being the familiar singular series associated with the representation of positive integers as sums of four cubes.

The following theorem is proved.

Theorem. Whenever \(192/869 \leq \theta \leq 2/9\), we have \(r_\theta(n) \geq 1\) for almost all integers \(n\).

An additional asymptotic formula is established.