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Claim. Corollary 2 of G. Tenenbaum, On block Behrend sequences, Math. Proc. Cambridge Philos. Soc. 120 (1996), no. 2, 355--367 (card tenenbaum_1996_block_behrend_sequences). A set $A$ of integers greater than $1$ is Behrend when its set of multiples $M_A$, in the notation of Problem 691, has asymptotic density $1$. Let $A=\bigcup_j(T_j,H_jT_j]\cap\mathbb Z^+$ be a block sequence, that is $1+T_j^{\eta-1}\le H_j\le\min\{T_j,T_{j+1}/T_j\}$ for a fixed $\eta>0$. If $1+c_1\le T_{j+1}/T_j\le1+c_2$ for positive constants $c_1,c_2$ and $H_j=1+j^{-\alpha}$ with $\alpha>0$, then $A$ is Behrend if $\alpha<\log2$ and is not Behrend if $\alpha>\log2$. The paper notes that Erdős's original claim, a critical exponent under the one-sided condition $T_{j+1}/T_j\ge1+c_1$ alone, is false as it stands: Theorem A of Hall and Tenenbaum makes $A$ non-Behrend for every $\alpha$ when, for instance, $T_j=\exp\exp j$. Tenenbaum writes that he understood from later discussions with Erdős that a two-sided condition on the ratios was meant, so that the corollary confirms the conjecture exactly, with the added information that the critical exponent is $\log2$. The paper presents the corollary as an immediate consequence of its Theorem 1 and of Theorem A. The page is dated to the issue month, August 1996, as the publisher's record gives it.

Covers. Erdős's threshold conjecture for this family of block sequences, in its two-sided form, proved with critical exponent $\log2$. The case $\alpha=\log2$ and the general question, a necessary and sufficient condition on an arbitrary $A$ for $M_A$ to have density $1$, remain open; the paper calls effective general criteria very difficult, if not hopeless, to obtain with present techniques.

Depends on. No page of this wiki. The corollary follows from the paper's Theorem 1 and from Theorem A of Hall and Tenenbaum, which the paper cites.

Acceptance. Refereed: the journal paper cited above. The site's commentary says that Tenenbaum proves this conjecture, but the site labels the problem OPEN, so the commentary is a remark on a partial result and adds no reviewed evidence.