A new proof of Sárközy's theorem (Q2839296)

\textit{A. Sárközy} [Acta Math. Acad. Sci. Hung. 31, 125--149 (1978; Zbl 0387.10033)] and \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] proved a conjecture of Lovász stating that a set of integers of positive density contains a difference being a (non-zero) square. The current record on quantitative bounds for the density \(D(N)/N\) of an extremal set in \([1,N]\) is by \textit{J. Pintz, W. L. Steiger} and \textit{E. Szemerédi} [J. Lond. Math. Soc., II. Ser. 37, No. 2, 219--231 (1988; Zbl 0651.10031)].NEWLINENEWLINEThe paper under review does not aim to improve this bound quantitatively, but gives a quite elementary and self contained proof of the original conjecture. The method of proof is based on an idea of \textit{E. Croot} and \textit{O. Sisask} [Proc. Am. Math. Soc. 137, No. 3, 805--809 (2009; Zbl 1202.11011)], giving complete details to the estimates required for the circle method setup. It compares the density \(D(N)/N\) of a maximal set, without a square difference, with \(D(M)/M\), where \(N \ge \exp( c M^7)\), for some \(c>0\).NEWLINENEWLINE The author proves the estimate \(\frac{D(N)}{N} \le \frac{3}{4} \frac{D(M)}{M}\), which implies \(D(N)=o(N)\).