On additive complements. III (Q402622)

Two sets \(A,B\) of non-negative integers are called ``additive complements'' if every sufficiently large positive integer is sum of two numbers, one of them belonging to \(A\) and the other one to \(B.\) Of course \(A\) and \(B\) may have elements in common or even be equal.NEWLINENEWLINEIt is known (see [\textit{J. Fang} and \textit{Y. Chen}, Proc. Am. Math. Soc. 138, No. 6, 1923--1927 (2010; Zbl 1202.11010)] for a reference) that the counting functions \(A(x)=|A\cap[1,x]|\) and \(B(x)\) of two additive complements \(A\) and \(B\) verify \(A(x)B(x)>x-c\) for a constant \(c\) and all \(x\geq 1\). Consequently NEWLINE\[NEWLINE\liminf_{x\to+\infty}x^{-1}A(x)B(x)\geq 1.NEWLINE\]NEWLINE \textit{A. Sárközy} and \textit{E. Szemerédi} [Acta Math. Hung. 64, No. 3, 237--245 (1994; Zbl 0816.11013)] proved in 1994 that ifNEWLINENEWLINE\noindent \(\limsup_{x\rightarrow+\infty}x^{-1}A(x)B(x)\leq1\) (that is, if \(\lim_{x\rightarrow+\infty}x^{-1}A(x)B(x)=1),\) then NEWLINE\[NEWLINE\lim_{x\to+\infty}A(x)B(x)-x=+\infty.\tag{1} NEWLINE\]NEWLINENEWLINENEWLINEThe authors of the paper under review proved in [Proc. Am. Math. Soc. 138, No. 6, 1923--1927 (2010; Zbl 1202.11010)] that (1) still holds when NEWLINE\[NEWLINE\limsup_{x\rightarrow+\infty}x^{-1}A(x)B(x)>2 ~~\text{or}~~\limsup_{x\rightarrow+\infty}x^{-1}A(x)B(x)<{5\over 4}.NEWLINE\]NEWLINE In the above result, the number 2 cannot be replaced by a smaller one: this was pointed out by the authors in [ibid. 139, No. 3, 881--883 (2011; Zbl 1202.11010)]. It was an open problem to replace \({5\over 4}\) by a bigger number. This is done in this paper: the authors managed to replace \({5\over 4}\) by \(3-\sqrt 3.\)NEWLINENEWLINENEWLINEThe paper is well written, contains a clear introduction to the subject and informs the reader about results on additive complements in the set of \textit{all} integers.NEWLINENEWLINENEWLINE\texttt{Open problems:} Is it possible to replace \(3-\sqrt 3\) by a bigger number? Possibly, find the biggest such number. Estimate the growth of \(A(x)B(x)-x\) when (1) holds.