On the density of certain sequences of integers

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A. S. Besicovitch, "On the density of certain sequences of integers," Mathematische Annalen 110(1), 336--341 (1935). https://doi.org/10.1007/BF01448032.

source card.

Write $M(B)$ for the set of positive multiples of a set $B$. The paper begins with the two questions suggested by primitive abundant numbers: must every primitive set have density zero, and must its set of multiples have a natural density? (Introduction, p. 336.) Its construction answers both questions negatively.

The input is the divisor-window estimate. For

$$E_i=M([2^i,2^{i+1}))$$

and $e_i=d(E_i)$, Theorem 1 proves $e_1+\cdots+e_l=o(l)$ (§5, pp. 339--340). The proof first removes the density-zero set of integers having abnormally many divisors and then counts, over a long factorial period, how many dyadic divisor windows the remaining integers can meet; equations (5)--(8), pp. 339--340, are the quantitative core. Consequently there are arbitrarily remote windows with $e_i$ as small as prescribed.

In §7 (pp. 340--341), choose $\varepsilon<1/4$ and positive $\varepsilon_k$ with

$$\sum_{k\geq1}\varepsilon_k<\frac{\varepsilon}{2},$$

then select $i_1<i_2<\cdots$ so that $e_{i_k}<\varepsilon_k$ and each new scale lies beyond a factorial period for the preceding window; the displayed choice on p. 340 is $2^{i_{k+1}}>(2^{i_k+1})!$. Put $T_k=2^{i_k}$ and define

$$G=\bigcup_{k\geq1} \left([T_k,2T_k)\setminus\bigcup_{j<k}E_{i_j}\right), \qquad H=\bigcup_{k\geq1}E_{i_k}=M(G).$$

This is the block/gliding mechanism. Each fresh block $[T_k,2T_k)$ is moved far enough out that the old periodic sets have settled to their small mean densities. Deleting the old multiples makes $G$ primitive: an earlier member cannot divide a later one, and a later member is too large to divide an earlier one. The deletion loses at most $2\sum_{j<k}\varepsilon_j$ of a fresh block. At the same time every integer in the whole fresh block belongs to $H$, because it is a multiple of itself; a deleted generator was already a multiple of an earlier block, which also explains $H=M(G)$.

The two cutoff subsequences force the failure of natural density. Immediately before a fresh block, at $T_k$, only the old multiple sets contribute, giving

$$\underline d(H)<2\varepsilon_1+2\varepsilon_2+\cdots<\varepsilon.$$

At the end of the block, $2T_k$, the interval $[T_k,2T_k)$ is contained in $H$, so

$$\overline d(H)>\frac12.$$

The paper's conclusions follow on p. 341, which the copy read lacks, so their printed form is not checked here; the construction also gives $\underline d(G)=0$ and $\overline d(G)>1/2-2\sum_k\varepsilon_k>1/4$. Thus $G$ refutes the proposed zero-density consequence of primitivity, while its multiple closure $H$ refutes natural-density existence for arbitrary sets of multiples.

For Problem 25, take the forbidden class $0\pmod g$ for each $g\in G$. The excluded set is exactly $H=M(G)$ and the survivor set is $\mathbb N\setminus H$, so Besicovitch supplies a clean model of how gliding blocks can make ordinary densities oscillate. It is not a near-counterexample to E0025, which asks for logarithmic density. The later Davenport--Erdős theorem says that every set of multiples has a logarithmic density (indeed equal to its lower natural density), so both $H$ and its complement have logarithmic densities despite the ordinary-density failure.

Reading status. Claims checked for Theorem 1 and the §7 construction against the page images of printed pp. 339--340. The scan read ends with the definition of $H$ on p. 340, so the density conclusions given above for p. 341 were not checked against it; no full proof verification was undertaken.

Results to transcribe.

• Theorem 1 (§5, pp. 339--340): for $E_i=M([2^i,2^{i+1}))$, one has $\sum_{i\leq l}d(E_i)=o(l)$.
• Theorem 2 (§6, p. 340): for the stated sequence $n_{i+1}=n_i^{1+\log^{-\alpha}n_i}$, $\log 2<\alpha<1$, the analogous divisor-window densities satisfy $m_1+\cdots+m_l=o(l)$.
• §7 construction and conclusion (pp. 340--341): there is a primitive set $G$ with lower density zero and upper density greater than $1/4$, and its set of multiples $H=M(G)$ has lower density below $\varepsilon<1/4$ and upper density above $1/2$, hence no natural density.