---

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 1 (p. 4). Let $\mathcal A=\bigcup_j(T_j,H_jT_j]\cap\mathbb Z^+$ be a block sequence such that, for real constants $\alpha,\gamma,\sigma,\tau$ with $\sigma>-1$,

$$\log(T_{j+1}/T_j)\asymp j^\sigma(\log2j)^\tau,\qquad \log H_j\asymp j^{-\alpha}(\log2j)^\gamma\qquad(j=1,2,\dots).$$

Put $\sigma_0:=\log2/(1-\log2)$ and

$$\alpha_0(\sigma):=\begin{cases}(1-\log2)(\sigma_0-\sigma)&\text{if }-1<\sigma\le\sigma_0,\\ \sigma_0-\sigma&\text{if }\sigma>\sigma_0.\end{cases}$$

Then $\mathcal A$ is a Behrend sequence if $\alpha<\alpha_0(\sigma)$, and is not a Behrend sequence if $\alpha>\alpha_0(\sigma)$.

The corollary says nothing about the boundary case $\alpha=\alpha_0(\sigma)$. The paper calls Corollary 2 a very special case of this one (p. 4); its hypotheses fall under $\sigma=\tau=\gamma=0$, where the threshold is $\alpha_0(0)=(1-\log2)\sigma_0=\log2$.

Source. G. Tenenbaum, On block Behrend sequences, Math. Proc. Cambridge Philos. Soc. 120 (1996), no. 2, 355--367, DOI 10.1017/S0305004100074910; Corollary 1 on p. 4. Page numbers are those of the author's typescript identified on the source card.

Read depth. Claims checked: the statement was read clause by clause on the page image of p. 4. The paper prints no proof, and none was worked out here.

Proof pointer

The paper calls Corollaries 1 and 2 immediate consequences of Theorems A and 1 and omits the verification (p. 4). Theorem 1 supplies Behrend sequences and Theorem A (Hall and Tenenbaum 1992, the paper's [7], stated on p. 2) supplies both the necessary condition (1·2) and, for stretched sequences, a sufficient one; the verification of which case applies across the stated ranges is not reconstructed here.

Dependencies

Theorem 1 and Theorem A (p. 2), the latter 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 $1$. The corollary decides the question for the block sequences it describes, except at the boundary exponent $\alpha=\alpha_0(\sigma)$; it does not give a criterion for general $A$.