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Statement

Setting (pp. 1--2). $A(x)$ and $B(x)$ count the elements up to $x$, condition (1.2) is $A(x)B(x)/x\to1$, and $a^*(x)=\max\{a\in A,\ a\le x\}$.

Theorem 1.3 (p. 2, quoted). "Let $\omega$ be a function tending to infinity arbitrarily slowly. There are additive complements satisfying (1.2) such that for infinitely many values of $x$ we have

$$A(x)B(x)-x<\min\bigl(\omega(x),ca^*(x)\bigr) \tag{1.7}$$

with some constant $c$."

The paper introduces the theorem (p. 2) as the answer to the question, which it says Chen and Fang also formulated, whether an absolute lower bound such as $A(x)B(x)-x>\log x$ holds: it does not. It is the example by which the abstract says the paper's lower bound, Theorem 1.2, is nearly best possible.

Source. I. Z. Ruzsa, Exact additive complements, Q. J. Math. 68 (2017), 227--235, doi:10.1093/qmath/haw029; labels and pages are those of the arXiv version arXiv:1510.00812v1 (3 October 2015), as identified on the source card: the statement on p. 2, the construction in Section 3, pp. 5--7.

Read depth. Claims checked: the statement was read clause by clause on the printed page, and the construction (pp. 5--7) was read through for structure. No step of it was independently checked, and nothing here is independently reviewed.

Proof pointer

Section 3, pp. 5--7. Take primes $p_k$ with $k^3<p_k<(k+1)^3$ (with finitely many exceptions) and a fast-growing sequence $u_k$ with $u_k>ku_{k-1}$ and $p_k\mid u_k$. Let $A$ be the union of blocks $A_1=\{1,\ldots,p_1\}$ and $A_k\subset(u_k,2u_k)$ of $p_k-p_{k-1}$ elements, chosen so that $A_1\cup\cdots\cup A_k$ is a complete residue system modulo $p_k$; Lemma 3.1 (p. 5) shows such blocks exist once $u_k$ exceeds a bound depending only on the primes. Let $B$ be the union of the sets $B_k$ of multiples of $p_k$ in $(ku_k,(k+3)u_{k+1})$. Every $n>3u_1$ lies in $A+B$ (p. 6), and counting gives $A(x)B(x)-x=O(x/k)$, so the complements are exact. At $x=u_{k+1}$ one has $A(x)=p_k$ and $u_k<a^*(x)<2u_k$, giving $A(x)B(x)-x<c_3u_k<c_3a^*(x)$, and $c_3u_k<\omega(x)$ once $u_j$ grows so fast that $\omega(u_{k+1})>u_k$ (p. 7).

Dependencies

Lemma 3.1 (p. 5) and the convergence of $\sum1/p_i$ over the chosen primes; no other result of the paper.

Bears on

• Problem 785: the sets constructed are infinite, $A+B$ contains every integer above $3u_1$, and $A(x)B(x)\sim x$, so they satisfy the problem's hypotheses (as sets of positive integers; this matching is an observation on this page, not the paper's). Along infinitely many $x$ their excess $A(x)B(x)-x$ stays below $\omega(x)$ for a prescribed $\omega$ tending to infinity arbitrarily slowly. So the excess in the problem's conclusion $A(x)B(x)-x\to\infty$ admits no absolute lower bound such as $\log x$, as the paper says (p. 2); the theorem does not contradict the conclusion.