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Statement

Block sequences and Behrend sequences are as in the setting of Theorem 1: $\mathcal A=\bigcup_j(T_j,H_jT_j]\cap\mathbb Z^+$ with $1+T_j^{\eta-1}\le H_j\le\min\{T_j,T_{j+1}/T_j\}$ for a fixed $\eta>0$, and $\mathcal A$ is Behrend when its set of multiples has asymptotic density $1$.

Corollary 2 (p. 5). Let $\mathcal A=\bigcup_j(T_j,H_jT_j]\cap\mathbb Z^+$ be a block sequence such that, for some positive constants $c_1,c_2$,

$$1+c_1\le T_{j+1}/T_j\le1+c_2\qquad(j=1,2,\dots),$$

and suppose moreover that $H_j=1+j^{-\alpha}$ for $j\ge1$, with $\alpha>0$. Then $\mathcal A$ is a Behrend sequence if $\alpha<\log2$, and is not a Behrend sequence if $\alpha>\log2$.

The corollary says nothing about $\alpha=\log2$. The paper calls it a very special case of Corollary 1 (p. 4): its hypotheses give $\log(T_{j+1}/T_j)\asymp1$ and $\log H_j\asymp j^{-\alpha}$, the case $\sigma=\tau=\gamma=0$ there, where $\alpha_0(0)=\log2$.

Erdős's conjecture (p. 5). The paper reports that Erdős's original claim was the existence of a critical value $\alpha_0\in(0,1)$ under the one-sided condition $T_{j+1}/T_j\ge1+c_1$ alone, and that this is false as it stands: by Theorem A the sequence is not Behrend for any $\alpha$ when, for instance, $T_j=\exp\exp j$. Having learned from Erdős that he intended a two-sided condition on the ratios, the paper says the corollary confirms his conjecture exactly, with $\alpha_0=\log2$.

Source. G. Tenenbaum, On block Behrend sequences, Math. Proc. Cambridge Philos. Soc. 120 (1996), no. 2, 355--367, DOI 10.1017/S0305004100074910; Corollary 2 and the discussion of Erdős's conjecture on p. 5. Page numbers are those of the author's typescript identified on the source card.

Read depth. Claims checked: the statement and the discussion following it were read clause by clause on the page image of p. 5. The paper prints no proof, and none was worked out here.

Proof pointer

The paper calls Corollaries 1 and 2 immediate consequences of Theorem 1 and Theorem A and omits the verification (p. 4); it also presents Corollary 2 as a very special case of Corollary 1. The verification is not reconstructed here.

Dependencies

Theorem 1, Corollary 1, and Theorem A (p. 2), from R. R. Hall and G. Tenenbaum, On Behrend sequences, Math. Proc. Cambridge Philos. Soc. 112 (1992), 467--482.

Bears on

• Problem 691: the problem asks for a necessary and sufficient condition for $M_A$ to have density $1$, and the problem page records Erdős's block example, intervals $(n_k,(1+\eta_k)n_k)$ with $\eta_k=k^{-\beta}$ and a conjectured threshold in $\beta$. Reading $n_j$ as $T_j$ and $\eta_j$ as $j^{-\alpha}$, with the paper's half-open blocks and its two-sided condition on the ratios, the corollary places the threshold at $\log2$. It does not decide $\alpha=\log2$, it does not cover the one-sided version (which the paper says fails as stated), and it gives no criterion for general $A$. The claim page records the result.