The number of squares and \(B_h[g]\) sets (Q2773298)

A \(B_h[g]\) set is a set \(A\) of integers with the property that every integer has at most \(g\) representations as a sum of \(h\) elements of \(A\). Let \(A(h,g,N)\) denote the size of the maximal such set \(A\subset [1,N]\). Until recently, estimates of these quantities were based exclusively on combinatorial counting arguments. \textit{J. Cilleruelo, I. Z. Ruzsa} and \textit{J. J. Trujillo} [J. Number Theory 97, No. 1, 26--34 (2002; Zbl 1041.11015)] introduced a Fourier-analytic technique to estimate \(A(2,g,N)\). This paper is a systematic study of Fourier-analytic methods from this aspect. The ``square'' in the title refers to quadruples of numbers in the form \(a,a+b,a+c,a+b+c\) (projections of a suitable square). The number of such squares is the fourth moment of the Fourier transform, and it is in close connection with the \(B_h[g]\) property. This connection is evident in the \(h=2\), \(g=1\) case (Sidon sets), and in the paper several indirect and nontrivial connections are found. NEWLINENEWLINENEWLINEAs a result, many of the known upper estimates for \(A(h,g,N)\) are improved. The reviewer feels, however, that the methods developed are more important than these numerical improvements and will certainly have further applications.