On Waring's problem for four cubes (Q1895998)

It is generally conjectured that the number of representations of a number \(n\) as a sum of four cubes is of the form \(C_n n^{1/3} + O(E(n))\), where \(C_n = \Gamma^3 ({4 \over 3}) {\mathfrak S} (n)\), \({\mathfrak S} (n)\) being the usual ``singular series'', and \(E(n) = n^{1/3} (\log n)^{- \rho}\) for some constant \(\rho > 0\). \textit{R. C. Vaughan} [J. Reine Angew. Math. 365, 122-170 (1986; Zbl 0574.10046)] showed that this formula holds for all but at most \(O(N (\log N)^{- \rho})\) integers \(n\) not exceeding \(N\). The authors show that the formula holds (with \(\rho = {1 \over 5})\) for all but \(O(M (\log N)^{- 1/4})\) integers in a short interval \(N < n \leq N + M\), where \(M = N^\theta\) and \({5 \over 6} < \theta < 1\). As usual, the Hardy-Littlewood method is used. The usual procedure of invoking a Bessel inequality on the minor arcs would be unsatisfactory in \(M\)-aspect, so the authors introduce a more sophisticated Fourier analysis leading to auxiliary equations in which the \(M\)-dependence is retained.