On arithmetic progressions in \(A+B+C\) (Q2927908)

In this paper the author investigates the additive structure of the three terms sumsets namely, long arithmetic progressions in the sumset \(A+B+C\), where \(A\), \(B\) and \(C\) are subsets of \(\mathbb{Z}_{N}\). He proves that If \(0 < \varepsilon < 1\) and \(A\), \(B\) and \(C\) are subsets of \(\mathbb{Z}_{N}\) with densities \(\alpha \geq (\log N)^{-2+\varepsilon}\) and \(\beta\), \(\gamma \geq e^{-c(\log N)^{c}}\) respectively, then there exists an absolute positive constant \(c\) such that the sumset \(A+B+C\) contains an arithmetic progression of length at least \(e^{c(\log N)^{c}}\). As a corollary he obtains that if \(A\) is a subset of primes less than \(N\) with cardinality \(\alpha N/\log N\) where \(\alpha \geq (\log N)^{-1}(\lg\log N)^{14}\), then there exists an absolute constant \(c > 0\) such that \(A+A+A\) contains an arithmetic progression of length at least \(e^{c(\alpha\log N)^{1/4}(\log\log N)^{-7/2}}\).NEWLINENEWLINEThe proofs are based on Fourier analytic techniques. The key tool is the well known density-increment argument. To use this he needs some properties of Bohr sets, the Katz-Koester transform established by \textit{T. Sanders} [Ann. Math. (2) 174, No. 1, 619--636 (2011; Zbl 1264.11004)] and the Croot-Sisask lemma [\textit{E. Croot} and \textit{O. Sisask}, Geom. Funct. Anal. 20, No. 6, 1367--1396 (2010; Zbl 1234.11013)] about sets of almost-periods of convolutions.