Additive representation in thin sequences. III: Asymptotic formulae (Q2773291)

This paper continues the series opened in [Ann. Sci. Éc. Norm. Supér., IV. Sér. 34, 471-501 (2001; Zbl 1020.11062)], to which readers are referred for an overview of the strategy followed. The aim is to show that an expected asymptotic formula holds for the number of representations of a number in a specified form, and this for almost all occurrences of the number in a given polynomial sequence. NEWLINENEWLINENEWLINESlightly more generally, let \(\phi\) be an integral polynomial, that is to say one such that \(\phi(t)\) is an integer whenever \(t\)~is an integer. Let \(R_s(n)\) denote the number of representations of~\(n\) as a sum of \(s\) positive cubes. Then one expects an asymptotic formula with a main term \(M_s(n) = f(s) {\mathfrak S}(n) \smash {n^{s/3-1}}\), where \({\mathfrak S}(n)\) is a singular series and \(f(s)\) is expressed in terms of \(\Gamma\)~functions. Let \({\mathcal E}_{s,\phi}(N,\gamma)\) denote the number of integers~\(n\) with \(1 \leq n \leq N\) for which \(\phi(n)>0\) and \(|R_s(n)-M_s(n)|> \smash{{\phi(n)}^{s/3-1}}/\log^\gamma n\). When \(\phi\) is a quadratic polynomial the authors show that NEWLINE\[NEWLINE{\mathcal E}_{6,\phi}(N,\gamma)\ll N/\log^\delta NNEWLINE\]NEWLINE whenever NEWLINE\[NEWLINE\delta<\max\{2-\gamma,5/2-2\gamma\}.NEWLINE\]NEWLINE For sums of seven cubes and quadratic \(\phi\) their result reads NEWLINE\[NEWLINE{\mathcal E}_{7,\phi}(N,\gamma)\ll N^{2/3+\delta}NEWLINE\]NEWLINE whenever \(\gamma>0\) and~\(\delta>0.\) For cubic \(\phi\) they show NEWLINE\[NEWLINE{\mathcal E}_{7,\phi}(N,\gamma)\ll N/\smash{\log^\delta N}NEWLINE\]NEWLINE whenever \(0<\delta<2-2\gamma<2\). NEWLINENEWLINENEWLINEThey give a number of other consequences of their methods, relating, for example, to \(\phi\) of higher degree and to sums of \(k\)th powers, possibly of primes. NEWLINENEWLINENEWLINEA previous instance of a result of this general type, involving thin sequences, appears in a paper by \textit{J. Brüdern} and \textit{N. Watt} [Duke Math. J. 77, 583-606 (1995; Zbl 0828.11051)] dealing with sums of four cubes in almost all intervals of a certain short length. As in the present instance, this paper uses the Hardy-Littlewood method, but had to deal with the difficulty that the usual procedure involving an appeal to Bessel's inequality is not suitable for problems involving thin sequences.