On the density of certain sequences of integers

../

construction_p340: States Besicovitch's construction from Theorem 1 of a primitive set G, built from remote dyadic blocks with their earlier multiples removed, whose set of multiples H oscillates between lower density at most 2 sum eps_k and upper density at least 1/2.

theorem_1: States Besicovitch's theorem that if e_i is the density of the integers with a divisor at least 2^i and below 2^(i+1), then e_1 + ... + e_l = o(l), so e_i is small for almost all i.

theorem_2: States Besicovitch's extension of Theorem 1 to the windows between n_i and n_(i+1) = n_i^(1 + (log n_i)^(-alpha)) with log 2 < alpha < 1: the densities m_i of the integers with a divisor in these windows satisfy m_1 + ... + m_l = o(l).

---

A. S. Besicovitch, "On the density of certain sequences of integers," Mathematische Annalen 110(1), 336--341 (1935). https://doi.org/10.1007/BF01448032. No notice is printed in the publisher's scan, which has no text layer (its first and last pages, printed pp. 335 and 340, read as page images); the publisher's article page (https://link.springer.com/article/10.1007/BF01448032, read 2026-10-02 through its cookie hop) offers the PDF behind a paywall with "Reprints and permissions", no Open Access or Creative Commons statement and no article-year copyright line, only the site footer "© 2026 Springer Nature", every other right reserved.

The copy read for this card is a six-page publisher's scan of the 1935 printing covering printed pp. 335--340 (p. 335 is the end of the preceding article); it lacks the concluding p. 341.

Write $M(B)$ for the set of positive multiples of a set $B$. The paper begins with the two problems of H. Davenport and S. Chowla 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 $2^{i_{k+1}}>2^{i_k+1}!$, printed without brackets and read here as $(2^{i_k+1})!$. Put $T_k=2^{i_k}$ and define

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

The print on p. 340 writes $H=E_1+E_2+E_3+\cdots$, evidently for $E_{i_1}+E_{i_2}+\cdots$, since the union of all the $E_i$ is every integer $n\geq2$.

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

$$\underline d(H)\leq2\varepsilon_1+2\varepsilon_2+\cdots<\varepsilon.$$

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

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

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 $\overline d(G)\geq1/2-\sum_k\varepsilon_k>3/8$. These density bounds are the card's own derivation from the construction, not the paper's printed statement. 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 [[integer_sequences/davenport_1936_sequences_positive_integers/_index|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 Theorems 1 and 2 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.

Bears on. #25: with the members of $G$ as moduli and residue class $0$ for each, the problem's set $A$ is $\mathbb N\setminus H$, which has no natural density; the problem asks about logarithmic density, which the construction does not address and which this $A$ has by the Davenport--Erdős theorem. #143: $G$ is a primitive set, so it satisfies the problem's hypothesis, and it has positive upper density, so the hypothesis does not force natural density zero; the construction does not address the series or the logarithmic density the problem asks about. #446: Theorem 1 gives $\liminf_n\delta(n)=0$ for the problem's $\delta(n)$, since $\delta(2^i)\leq e_i$; it gives neither $\delta(n)\to0$ nor the growth rate the problem asks for.

Results. Theorem 1 (§5, p. 339, proof pp. 339--340): $e_1+\cdots+e_l=o(l)$ for the dyadic divisor-window densities $e_i$. Theorem 2 (§6, p. 340): for $\log2<\alpha<1$ and $n_{i+1}=n_i^{1+\log^{-\alpha}n_i}$, the densities $m_i$ of the integers with a divisor $\geq n_i$ and $<n_{i+1}$ satisfy $m_1+\cdots+m_l=o(l)$; no proof is printed. The §7 construction (p. 340): the primitive set $G$ and its set of multiples $H$, with the density properties derived on that page; the paper's own conclusion on p. 341 is not in the copy read. Lemmas 1--3 (pp. 337--338) are proof steps of Theorem 1, summarized on its page.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.