On additive complements. IV (Q2299624)

Let \(A,B\) be infinite sequences of non-negative integers and let \(A(x),B(x)\) be the corresponding counting functions. Denote by \(r(x)\) the numbers of integers \(\le x\) not of the form \(a_i+b_j\) (\(a_i\in A, b_j\in B\)) and by \(a^*(x)\) the largest element of \(A\) in \([1,x]\). In a previous paper [Combinatorica 39, No. 4, 813--823 (2019; Zbl 1431.11019)] the authors have shown that if \(r(x)=o(a^*(x))\) and one has \[ \lim_{x\to\infty}\frac{A(2x)}{A(x)}=1\quad \text{or} \quad\lim_{x\to\infty}\frac{B(2x)}{B(x)}=1,\tag{1} \] then \[ A(x)B(x)-x+r(x)\ge\left(1-o(1)\right)\frac{a^*(x)}{A(x)} \] holds for \(x\) tending to infinity. Now they prove that the assumption (1) may be omitted and establish the more explicit inequality \[ A(x)B(x)-x+r(x)\ge \left(1-2M(1-\frac1\delta)-o(1)\right)\frac{a^*(x)}{A(x)}, \] where \[ \delta=\limsup_{x\to\infty}\frac{A(2x)}{A(x)},\quad M=\min\left\{\limsup_{x\to\infty}\frac{A(x)B(x)}x,2\right\}. \] This implies (Corollary 1.3) that if \(A,B\) are infinite additive complements, i.e. all sufficiently large integers are of the form \(a_i+b_j\), then the inequality \[ \limsup_{x\to\infty}\frac{A(2x)}{A(x)}<\frac43\tag{2} \] implies \[ A(x)B(x) - x\to\infty. \] In the case when in (2) ``\(<4/3\)'' is replaced by ``\(=1\)'' this has been established by \textit{A. Sárközy} and \textit{E. Szemerédi} [Acta Math. Hung. 64, No. 3, 237--245 (1994; Zbl 0816.11013)], confirming a conjecture of \textit{L. Danzer} [J. Reine Angew. Math. 214/215, 392--394 (1964; Zbl 0126.27902)].