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Source. G. Tenenbaum, Some of Erdős' unconventional problems in number theory, thirty-four years later, in L. Lovász, I. Z. Ruzsa and V. T. Sós (eds), Erdős Centennial, Bolyai Society Mathematical Studies 25 (2013), 651--681. Labels and pages here are those of the author's version identified on the source card, paginated 1--22; the published chapter was not read. Equations (25), (26) and (27) are all on p. 15.

Read depth. Claims checked: the three statements were read clause by clause on the printed page. (25) and (26) are reported from other works; the step to (27) is stated in one sentence and not written out.

Statement

Notation (p. 2): $\mathcal M(\mathcal A)$ is the set of multiples of $\mathcal A$, and $\mathrm d$, $\underline{\mathrm d}$ are natural and lower density. A Behrend sequence is an integer sequence with $\mathrm d\mathcal M(\mathcal A)=1$ (pp. 13--14).

Equation (25) (p. 15), from Davenport and Erdős, On sequences of positive integers, J. Indian Math. Soc. 15 (1951), 19--24 (card):

$$\underline{\mathrm d}\mathcal M(\mathcal A) =\lim_{T\to\infty}\mathrm d\mathcal M(\mathcal A\cap[1,T]).\qquad(25)$$

The survey calls the right-hand side the sequential density of $\mathcal M(\mathcal A)$.

Equation (26) (p. 15). From (25) and Behrend's inequality for finite sequences, for all integer sequences $\mathcal A$, $\mathcal B$,

$$1-\underline{\mathrm d}\mathcal M(\mathcal A\cup\mathcal B) \ge\bigl\{1-\underline{\mathrm d}\mathcal M(\mathcal A)\bigr\} \bigl\{1-\underline{\mathrm d}\mathcal M(\mathcal B)\bigr\}.\qquad(26)$$

A footnote records an improvement by Ahlswede and Khachatrian (J. Number Theory 55 (1995), 170--180).

Equation (27) (p. 15). It follows, the survey says, that

$$\sum_{a\in\mathcal A}\frac1a=\infty\qquad(27)$$

is a necessary condition for $\mathcal A$ to be a Behrend sequence, and that every tail $\mathcal A\smallsetminus[1,T]$ of a Behrend sequence is again a Behrend sequence.

As printed, (27) carries no hypothesis on $\mathcal A$. It needs $1\notin\mathcal A$: the sequence $\{1\}$ is a Behrend sequence with reciprocal sum $1$ and no tail of it is one, and (26) gives no information when one of the two sequences contains $1$ (a remark of this page).

The survey adds (p. 15) that if $\mathcal A$ is a Behrend sequence, the number of divisors of $n$ in $\mathcal A$ tends to infinity for almost all $n$, a result of Hall and Tenenbaum (Math. Proc. Cambridge Philos. Soc. 112 (1992), 467--482).

Proof pointer

p. 15, where the step is one sentence; it is sketched here. Apply (26) to $\mathcal A\cap[1,T]$ and the tail $\mathcal A\smallsetminus[1,T]$; a finite sequence of integers greater than $1$ has $\mathrm d\mathcal M<1$, so the tail must have lower density of multiples $1$, and a tail of convergent reciprocal sum has density of multiples at most that sum, which tends to $0$.

Dependencies

(25), from Davenport and Erdős 1951; Behrend's inequality for finite sequences; neither is proved in the survey.

Bears on

• Problem 26: the survey does not state the problem. By (27), for an infinite set $A$ of positive integers with $\sum_{a\in A}1/a<\infty$, no shift $A+k$ with $k\ge1$ (whose elements all exceed $1$ and whose reciprocal sum also converges) is a Behrend sequence, so such an $A$ answers the problem's question in the negative. The survey does not draw this conclusion.
• Problem 691: (27) is a necessary condition for a Behrend sequence, not a sufficient one; the survey states the problem in Erdős's words (p. 13) and calls an effective general criterion seemingly hopeless (p. 15).