A sequence $A$ for which $M_A$ has density $1$ is called a Behrend sequence.

If $A$ is a set of prime numbers (or, more generally, a set of pairwise coprime integers without $1$) then a necessary and sufficient condition is that $\sum_{p\in A}\frac{1}{p}=\infty$.

The general situation is more complicated. For example suppose $A$ is the union of $(n_k,(1+\eta_k)n_k)\cap \mathbb{Z}$ where $1\leq n_1<n_2<\cdots$ is a lacunary sequence. (This construction is sometimes called a block sequence.) If $\sum \eta_k<\infty$ then the density of $M_A$ exists and is $<1$. If $\eta_k=1/k$, so $\sum \eta_k=\infty$, then the density exists and is $<1$.

Erdős writes it 'seems certain' that there is some threshold $\alpha\in (0,1)$ such that, if $\eta_k=k^{-\beta}$, then the density of $M_A$ is $1$ if $\beta <\alpha$ and the density is $<1$ if $\beta >\alpha$.

Tenenbaum notes in [Te96] that this is certainly not true as written since if the $n_j$ grow sufficiently quickly then this sequence is never Behrend, for any choice of $\eta_k$. He then writes 'we understand from subsequent discussions with Erdős that he had actually in mind a two-sided condition on' $n_{j+1}/n_j$.

Tenenbaum [Te96] proves this conjecture: if there are constants $1<C_1<C_2$ such that $C_1<n_{i+1}/n_i<C_2$ for all $i$ and $\eta_k=k^{-\beta}$ then $A$ is Behrend if $\beta<\log 2$ and not Behrend if $\beta>\log 2$.

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This page was last edited 28 December 2025.