Higher-order Fourier analysis of multiplicative functions and applications (Q2826780)

The aim of this article is the study of asymptotic behavior of averages of the following form NEWLINE\[NEWLINE{1\over N^2} \sum_{1\leq m,n\leq N}\, \prod^s_{i=1} f(L_i(m,n)),NEWLINE\]NEWLINE where \(f\) is an arbitrary multiplicative function of modulus at most 1, and \(L_i(m,n)\), \(i=1,2,\dots, s\), are linear forms with integer coefficients. The authors are mainly motivated by applications, one of them being that there is a link between this problem and partition regularity problems of nonlinear homogeneous equations of three variables.NEWLINENEWLINE In this way, the given methods enable to attack some previously intractable problems. The exact statements are too long and complicated to be stated here.