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Statement

Setting (p. 340, §7). The paper applies Theorem 1 to the two problems of H. Davenport and S. Chowla posed in its introduction (p. 336): whether a sequence no term of which divides another must have density zero, and whether the set of multiples of such a sequence must have a density.

Construction (p. 340). Take positive numbers $\varepsilon$ and $\varepsilon_k$ ($k=1,2,\ldots$) with

$$\varepsilon<\frac14,\qquad\sum_{k\geq1}\varepsilon_k<\frac{\varepsilon}{2}.$$

Let $E_i$ be the set of positive integers having a divisor $\geq2^i$ and $<2^{i+1}$, and $e_i$ its density. The paper observes that the mean density of $E_i$ on any interval of more than $2^i!$ consecutive integers is $<2e_i$. Using Theorem 1, choose integers $i_1<i_2<\cdots$ with $e_{i_k}<\varepsilon_k$ for every $k$ and, for $k\geq1$, $2^{i_{k+1}}> 2^{i_k+1}!$ (printed without brackets; read here as $(2^{i_k+1})!$). With $T_k=2^{i_k}$, the paper's set $G$ is, in this page's notation,

$$G=\bigcup_{k\geq1}\Bigl([T_k,2T_k)\setminus\bigcup_{j<k}E_{i_j}\Bigr),$$

its first block being all of $[T_1,2T_1)$. The paper's second set is printed as $H=E_1+E_2+E_3+\cdots$; the union of all the $E_i$ is every integer $n\geq2$, so the intended set is read here as $H=E_{i_1}\cup E_{i_2}\cup\cdots$.

The paper's statement of what the construction proves follows on p. 341, which the copy read for this source lacks (see the source card); its printed form is not recorded here.

Properties (observations of this page, derived from the construction above, not the paper's printed statement).

1. $G$ is primitive: within a block $[T_k,2T_k)$ no member divides another; a multiple in block $k$ of a member of block $j<k$ lies in $E_{i_j}$ and was removed; and a member of a later block exceeds every member of an earlier one.
2. $H$ is the set of multiples of $G$: each $n\in E_{i_k}$ has a divisor in $[T_k,2T_k)$, which is either in $G$ or in an earlier $E_{i_j}$, and induction on $k$ finishes.
3. $\underline d(H)\leq2\sum_k\varepsilon_k<\varepsilon$: below $T_k$ only $E_{i_1},\ldots,E_{i_{k-1}}$ meet $H$, and the observation on mean density applies to $[1,T_k)$.
4. $\overline d(H)\geq\frac12$, since $[T_k,2T_k)\subseteq H$ for every $k$. With item 3, $H$ has no natural density.
5. $\underline d(G)=0$, since $G\cap[1,T_k)\subseteq[1,2T_{k-1})$ and $T_k/T_{k-1}\to\infty$; and $\overline d(G)\geq\frac12-\sum_k\varepsilon_k>\frac38$, since the removed part of $[T_k,2T_k)$ has fewer than $2T_k\sum_{j<k}\varepsilon_j$ members.

So $G$ is a primitive set without density, and its set of multiples $H$ has no density, which answers both of the Davenport--Chowla questions in the negative.

Source. A. S. Besicovitch, "On the density of certain sequences of integers," Mathematische Annalen 110 (1935), 336--341, https://doi.org/10.1007/BF01448032: the Davenport--Chowla problems on p. 336, the construction of §7 on p. 340, its conclusion on p. 341.

Read depth. Claims checked: the construction was read clause by clause on p. 340. The paper's conclusion on p. 341 was not read; the properties above are this page's own derivation and are not independently reviewed.

Dependencies

Theorem 1 of the same paper, used only to pick windows with $e_{i_k}<\varepsilon_k$.

Bears on

• Problem 25: taking the moduli to be the members of $G$ in increasing order, each with residue class $0$, the problem's excluded set is $H$ and its set $A$ is $\mathbb N\setminus H$, which by items 3 and 4 has no natural density. The problem asks about logarithmic density, and the construction says nothing about it; by the theorem of Davenport and Erdős (source card) every set of multiples has a logarithmic density, so this $A$ has one.
• Problem 143: a primitive set of integers above $1$ satisfies the problem's hypothesis $\lvert kx-y\rvert\geq1$, and $G$ has positive upper density by item 5, so the hypothesis does not force natural density zero. The construction does not bear on the convergence of $\sum1/(x\log x)$ or on the logarithmic density the problem asks about.