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Statement

Setting (pp. 1--2). $\mathcal A$ is a strictly increasing sequence of integers exceeding $1$, $\mathcal M(\mathcal A)=\{ma:a\in\mathcal A,\ m\ge1\}$ is its set of multiples, and $\mathcal A$ is a Behrend sequence when $\mathcal M(\mathcal A)$ has asymptotic density $1$. A block sequence is a union $\mathcal A=\bigcup_{j\ge1}\mathcal A_j$ with $\mathcal A_j=(T_j,H_jT_j]\cap\mathbb Z^+$, where for some fixed $\eta>0$

$$1+T_j^{\eta-1}\le H_j\le\min\{T_j,\ T_{j+1}/T_j\}\qquad(j=1,2,\dots).$$

Theorem 1 (p. 3). Let $\mathcal A=\bigcup_{j\ge1}(T_j,H_jT_j]$ be a block sequence, and suppose that for some $\beta>1-\log2$ it satisfies the following five conditions.

1. (i) $T_{j+1}<T_j^2$ for $j=1,2,\dots$.
2. (ii) $\log H_j\asymp\log H_i$ whenever $T_i\le T_j\le T_i^2$.
3. (iii) $\log(T_{j+1}/T_j)\asymp\log(T_{i+1}/T_i)$ whenever $T_i\le T_j\le T_i^2$.
4. (iv) There is a $\varrho$ with $0<\varrho<\min\{\tfrac35,\tfrac32(\beta-1+\log2)\}$ such that $H_j>1+\exp\{-(\log T_j)^\varrho\}$ for $j=1,2,\dots$.
5. (v) The series diverges:

$$\sum_{j=1}^\infty\frac{\log H_j}{1+\log H_j} \Bigl(\frac{1+\log H_j}{\log T_j}\Bigr)^\beta=\infty.$$

Then $\mathcal A$ is a Behrend sequence.

Relation to the necessary condition (pp. 2 and 4). Theorem A of the paper, due to Hall and Tenenbaum (Math. Proc. Cambridge Philos. Soc. 112 (1992), 467--482, the paper's [7]), puts $\delta:=1-(1+\log_22)/\log2\approx0.08607$ ($\log_2$ the iterated logarithm) and takes $\beta<1-\log2$; for a block sequence that is sawn with respect to a function $\xi(j)\to\infty$, meaning every block has $\log H_j\le(\log T_j)/(\log_2T_j)^{\xi(j)}$ (1·1), being Behrend requires the series of (v) with this $\beta$ to diverge (1·2); for a stretched sequence the exponent is $\delta$. The paper remarks (p. 4) that, apart from the possibility of taking $\beta=1-\log2$, condition (v) coincides for sawn sequences with the necessary condition (1·2), so the theorem is essentially sharp, and it conjectures that the conclusion still holds with $\beta=1-\log2$. It also notes (p. 4) that conditions (i)--(iii) hold in most natural instances, while (iv) excludes very short blocks such as $H_j\le1+T_j^{c-1}$, a limitation of the method.

Source. G. Tenenbaum, On block Behrend sequences, Math. Proc. Cambridge Philos. Soc. 120 (1996), no. 2, 355--367, DOI 10.1017/S0305004100074910; Theorem 1 on p. 3, the remarks on pp. 3--4, the lemmas on pp. 6--12 and the proof in section 3, pp. 12--15. 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. 3. The lemmas and the proof (pp. 6--15) were read for structure only; no step was checked. Nothing here is independently reviewed.

Proof pointer

Pages 6--15, by the method of Maier and Tenenbaum (the paper's [8], also chapter 5 of Hall and Tenenbaum's Divisors). For an integer $n$ the proof works with $n_k$, the product of the distinct prime factors of $n$ up to $\exp\exp k$ (1·5), and bounds the proportion of $n\le x$ for which no $n_k$ has a divisor in a block. Section 2 (pp. 6--12) gives five lemmas: Lemma 1 (p. 6) on the number of prime factors of $n_k$ in ranges, Lemma 2 (p. 7) bounding a mean square of the exponential sum $\sum_jT_j^{i\vartheta}$ over the blocks with $J_1(k)<j\le J_2(k)$, Lemma 3 (pp. 8--9) bounding from below, through a weighted mean square of that sum, the Lebesgue measure $\lambda_k(m)$ of the set of reals $z$ for which $e^zd$ lies in one of those blocks $(T_j,H_jT_j]$ for some divisor $d$ of $m$ (in a slightly shrunk form), Lemma 4 (p. 9) a mean value bound involving $|\zeta(1+i\vartheta)|$, and Lemma 5 (p. 9) a lower bound for $\lambda_k(n_k)$ outside a small exceptional set. Section 3 (pp. 12--15) shows that the count $N_k$ of those $n\le x$ with no divisor of $n_k$ in a block decreases by a factor $1-c\,R^{-4-\beta}\gamma_k^*$ every few steps of $k$ (here $R\ge1$ is a large fixed parameter and $\gamma_k^*$ a normalized local sum of the terms of (v)); condition (v) makes the sum of the $\gamma_k^*$ diverge, which brings the count below $2\eta x$ with $\eta\to0$ as $R\to\infty$.

Dependencies

Lemmas 1--5 of the paper (pp. 6--12). External inputs named in the proof: lemma 51.2, theorem 01, theorem 07 and lemma 30.1 of Hall and Tenenbaum, Divisors (Cambridge University Press, 1988; the paper's [6]); the prime number theorem in a strong form, with remainder $\ll t\exp\{-(\log t)^{a_1}\}$ for some $a_1>a$, where $a<\tfrac35$ is the parameter of Lemma 4 (p. 9); Vinogradov's bound $|\zeta(1+i\vartheta)|\ll_b1+(\log\vartheta)^b$ for $\vartheta>1$ and $\tfrac23<b<1$ (p. 10); and a sieve bound (Halberstam and Richert, Sieve methods, theorem 3.5, the paper's [5], or the elementary estimate (3·4)).

Bears on

• Problem 691: the problem asks for a necessary and sufficient condition for $M_A$ to have density $1$. The theorem is a sufficient condition for one class of $A$, block sequences meeting (i)--(iv), adjacent to the necessary condition (1·2) of Hall and Tenenbaum for sawn sequences; together with Theorem A it yields Corollary 2, the case the problem page records. It does not give a criterion for general $A$.