Additive representation in thin sequences. I: Waring's problem for cubes (Q2757189)

This paper is the first of a series in which the authors consider the representations of numbers lying in a thin sequence (such as the sequence of squares) in one of the forms normally arising in applications of the Hardy-Littlewood method, in this instance as a sum of cubes. Other applications of their ideas are to appear, or have appeared, in [Mathematika 47, 117--125 (2000; Zbl 1023.11053), Acta Arith. 100, 267--289 (2001; Zbl 0995.11055), Q. J. Math. 52, 423--436 (2001; Zbl 1037.11062) and Glasg. Math. J. 44, 419--434 (2002; Zbl 1046.11067)]. NEWLINENEWLINENEWLINELet \(E_s(N)\) denote the number of ``exceptional'' positive integers not exceeding~\(N\) which are not representable as a sum of \(s\)~positive integral cubes. The authors remark that methods developed to deal with the case \(s=4\) would show, for example, that \(E_6(N) \ll N^{23/42}\), but that this argument would fail to show that the majority of integer squares are representable as a sum of six positive cubes. NEWLINENEWLINENEWLINELet \(\phi(n)\) denote an integer polynomial (integer valued whenever \(N\)~is an integer), and let \(E_\phi(N)\) denote the number of positive integers~\(n \leq N\) for which \(\phi(n)\) is not the sum of six cubes. When \(\phi\) is quadratic with positive leading coefficient the authors' result is that \(E_\phi(N) \ll N^{19/28}\). When \(\phi\) is cubic their result is \(E_\phi(N) \ll N^{255/274}\). A slight variant of the argument used here shows that almost all positive integers of the form \(N - t^3\) are sums of six cubes. Thus in the ``seven cubes'' theorem one can almost prescribe the value of one of the seven cubes used to represent a specified number. NEWLINENEWLINENEWLINEThe authors provide an overview of the strategy adopted in this and subsequent papers of the series. In general one might define~\(r(n)\) as the number of representations of \(n\) as \(\smash{\sum_{1 \leq i \leq s}}a_i\) with \(a_i\) in specified subsets~\({\mathcal A}_i\) of~\([1,X]\). In a classical approach one deals with \(r_{\mathfrak C}(n)= \int_{\mathfrak C}\smash{\prod_{1 \leq i \leq s}}f_i(\alpha)e(-\alpha n) d\alpha\), where \(f_i(\alpha) = \sum_{x_i \in {\mathcal A}_i} e(\alpha x)\), and the contribution from minor arcs \(\mathfrak m\) is estimated via Bessel's inequality \(\sum_{n \in \mathbb Z} |r_{\mathfrak m}(n)|^2 \leq \int_{\mathfrak m}|\smash{\prod_{1 \leq i \leq s}}f_i(\alpha)|^2 d\alpha\). However, when \(n\)~is restricted to a thin subset~\(\mathcal B\) of \([X/2,X]\) this approach does not take significant advantage of the structure of~\(\mathcal B\). NEWLINENEWLINENEWLINEThe authors make a direct investigation of the exceptional set~\(\mathcal E\) of integers \(n\) in \(\mathcal B\) having no representations of the specified form, for which \(\sum_{n \in \mathcal E} \smash{\int_0^1} \smash{\prod_{1 \leq i \leq s}}f_i(\alpha)e(-\alpha n)d\alpha = 0\). The problem is then reduced to an estimation of a certain integral over minor arcs, \(\int_{\mathfrak m} \smash{\prod_{1 \leq i \leq s}}f_i(\alpha) K(-\alpha) d\alpha\), where \(K(\alpha) = \sum_{n \in \mathcal E} e(\alpha n)\). An application of Schwarz's inequality to this integral would recover the classical approach, but other lines of attack are now possible. For example, an application of Hölder's inequality could take advantage of information available for the fourth-power moment \(\smash{\int_0^1}|K(\alpha)|^4 d\alpha\). Other variants of this approach are promised for subsequent papers in the series.