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Statement

Setting (p. 109). $a_1<a_2<\cdots$ is any sequence of integers, and $b_1<b_2<\cdots$ are the integers divisible by none of the $a$'s.

Gap densities. Assume the density of the $b$'s exists, which the paper notes is certainly the case when $\sum1/a_i<\infty$. Then the density of the $b_i$ with $b_{i+1}-b_i=t$ exists (printed $b_{i-1}-b_i$ [sic]). This is the paper's generalization of Lemma 1; it gives no proof and remarks only that the statement follows almost immediately from a theorem of Davenport and Erdős: if $c_k$ is the density of the integers divisible by none of $a_1,\ldots,a_k$ and $c$ the density of those divisible by no $a_i$, then $c=\lim_{k\to\infty}c_k$.

First-moment tail. By the same theorem, the paper says, it is easy to see that if the density of the $b$'s exists and is positive, then to every $\varepsilon$ there is a $c_\varepsilon$ with

$$\sum_{\substack{b_{i+1}\le x\\ b_{i+1}-b_i>c_\varepsilon}}(b_{i+1}-b_i)<\varepsilon x.$$

The example. No stronger result holds in general, even when $\sum1/a_i<\infty$: taking the $a$'s to be the integers in the intervals $[2^k,2^k(1+1/k^2)]$ gives $\sum1/a_i<\infty$ but

$$\lim_{x\to\infty}\frac1x\sum_{b_i<x}(b_{i+1}-b_i)^{1+\varepsilon}=\infty,$$

with $\lim$ as printed; the exponent is faint in the print and reads as $1+\varepsilon$. Erdős adds that he does not know whether this can happen when $\sum1/a_i<\infty$ and $(a_i,a_j)=1$.

Source. P. Erdős, Some problems and results in elementary number theory, Publ. Math. Debrecen 2 (1951), 103--109, doi:10.5486/pmd.1951.2.2.04: the closing paragraphs on p. 109. The Davenport--Erdős theorem is cited there (footnote 4) from H. Davenport and P. Erdős, On sequences of integers, Acta Arithmetica 2 (1936), 147--151, and On sequences of positive integers, J. Indian Math. Soc. 15, Part A (1951), 19--24. The edition read is identified on the source card.

Read depth. Claims checked: the statements were read clause by clause on the printed page. The paper proves none of them, and nothing was checked beyond the statements.

Proof pointer

None in the paper beyond the reduction to the Davenport--Erdős theorem named above.

Dependencies

The Davenport--Erdős theorem, carded at davenport_1936_sequences_positive_integers and davenport_1951_sequences_positive_integers.

Bears on

• Problem 489: context for the general question. The remarks concern the sequence $B$ of the problem for arbitrary $A$, but control only the first moment; the example has $\sum1/a_i<\infty$ and an unbounded $(1+\varepsilon)$-moment, yet its $a$'s up to $x$ number far more than $x^{1/2}$ (an observation of this page, not of the paper), so it lies outside the problem's hypothesis $|A\cap[1,x]|=o(x^{1/2})$. The remarks settle no case of the problem beyond the squarefree one on the page for (23) with α = 2.