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Statement

The companion question (p. 309). Let $b_1<b_2<\cdots$ be an infinite sequence of integers with $B(x)>c_3\log x$ for every $x$. Is there then a sequence $A$ satisfying (1) of the main theorem, that is $A(x)<c_1x/\log x$ for every $x\ge1$, such that every $n$ can be written as $a_i+b_j$?

Announced theorem (p. 309, unnumbered, quoted in part). The author states that he proved, using a result of Erdős's paper (the paper's reference [1]), that there is a sequence $b_1<b_2<\cdots$ with $B(x)>c_3\log x$ such that if every $n$ can be written as $a_i+b_j$, then for infinitely many $x$

$$\text{"}A(x) > c_4\log\log x/\log x\text{ [sic]."}$$

The paper adds that in view of a result of Lorentz (reference [2]) this is best possible, and that it settles the question in the negative.

As printed, the bound lacks a factor $x$: the counting function of an infinite sequence eventually exceeds $c_4\log\log x/\log x$, so the printed inequality would hold for every such $A$ and could not settle the question. The reading $A(x)>c_4x\log\log x/\log x$ is the one consistent with the paper's two claims, since it is incompatible with (1) and Lorentz's theorem gives every sequence with $B(x)>c_3\log x$ a complement with $A(x)=O(x\log\log x/\log x)$. That reading is an inference from the context; the print does not state it.

The constant $c_3$ here reuses the symbol the paper also uses, a few lines earlier, for the bound on the number of representations.

Source. I. Ruzsa, Jr., On a problem of P. Erdős, Canad. Math. Bull. 15 (1972), no. 2, 309--310, doi:10.4153/CMB-1972-058-2; p. 309. The edition read is identified on the source card.

Read depth. Claims checked: the question and the announced statement were read clause by clause on the printed page. The paper gives no proof, so no proof was checked. Nothing here is independently reviewed.

Proof pointer

None: the paper announces the result and says the author will return to it on another occasion.

Dependencies

None in the corpus. External inputs named by the paper: a result of Erdős, Some results on additive number theory, Proc. Amer. Math. Soc. 5 (1954), 847--853, and, for sharpness, Lorentz, On a problem of additive number theory, Proc. Amer. Math. Soc. 5 (1954), 838--841 (the paper's reference list prints the pages as 838--891).

Bears on

No numbered problem directly. The announced result concerns complements of general sequences with $B(x)>c_3\log x$, not the powers of $2$ of Problem 221.