Congruence properties of the \(\Omega\)-function on sumsets
Let \({\mathcal A},{\mathcal B}\) denote subsets of \(\{1,2,\dots, N\}\) where \(N\) is sufficiently large in the context of the problems described below. In [Stud. Sci. Math. Hung. 23, 161-168 (1988; Zbl 0657.10053)]. \textit{A. Sárközy} proved that if \({\mathcal A},{\mathcal B}\) have more than \(N(\log N)^{-H}\) elements, where \(H>0\), then, for \(h=0,1\) there exist \(a\in {\mathcal A}\), \(b\in{\mathcal B}\) such that \(\Omega(a+b)\equiv h\pmod 2\). The object of the present paper is to extend and explore results of this type. Let \(k>0\) and \(h\) be integers; the authors show that if \({\mathcal A},{\mathcal B}\) are sufficiently dense subsets, then the number of pairs \(a,b\) with \(a\in{\mathcal A}\), \(b\in{\mathcal B}\) and \(\Omega(a+b)\equiv h\pmod k\) is approximately \(\frac{|{\mathcal A}| |{\mathcal B}|}{k}\). In particular, there exists such a pair if \[ (|{\mathcal A}| |{\mathcal B}|)^{1/2}< ckN(\log N)^{-E}, \] where \(c>0\) and \(E= E(k)>0\) are computable. They continue their investigation by considering when \(a+b\) \((a\in{\mathcal A}\), \(b\in{\mathcal B})\) is always a square, including the special case \({\mathcal A}={\mathcal B}\). Suppose \({\mathcal A},{\mathcal B}\) are sets with \(|{\mathcal A}|\geq|{\mathcal B}|\geq 2\) such that \(a+b\) is a square for all \(a\in{\mathcal A}\), \(b\in{\mathcal B}\); then Theorem 4 states that, for \(\varepsilon>0\), the greatest value of \(|{\mathcal A}|\) lies between the quantities \(\exp((\log 2\pm\varepsilon) \frac{\log N}{\log\log N})\). In contrast, it is also shown that if \({\mathcal A}\) is a set such that \(a+a'\) is square whenever \(a\neq a'\), \(a,a'\in{\mathcal A}\), then \(|{\mathcal A}|< 37\log N\).
scientific article; zbMATH DE number 1274756