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Claim. Let $z_1<z_2<\cdots$ be a real sequence with $z_{j+1}/z_j\to1$ such that the number of $z_j<N$ is $\ll N^{2-\delta}$ for some fixed $\delta>0$. Then for almost all $\alpha>0$ the sequence $\alpha,2\alpha,3\alpha,\ldots$ is uniformly distributed relative to $\{z_j\}$: the position of $n\alpha$ within the gap $[z_j,z_{j+1})$ that contains it, scaled to $[0,1)$, is uniformly distributed. This is the deduction (9) that follows the Theorem of Davenport and Erdős (p. 4). The Theorem itself counts the multiples of $\alpha$ that fall into a sparse union of non-overlapping intervals $(x_j,y_j)$: writing $I(Z)$ for the total length of the intervals starting below $Z$ and $F_\alpha(N)$ for the number of $n\le N$ with $n\alpha$ in the union, if $I(Z)\gg Z$ and at most $\ll N^{2-\delta}$ intervals start below $N$, then $\alpha F_\alpha(N)/I(N\alpha)\to1$ for almost all $\alpha>0$; taking the lower $\lambda$-parts of the gaps of $\{z_j\}$ as the intervals, for each $0<\lambda<1$, gives (9). No monotonicity of the gaps is assumed. The counting condition is the case $a_n\gg n^{1/2+\epsilon}$ in which the site's commentary and Schmidt's introduction state the theorem (at most $CN^{2-\delta}$ terms below $N$ gives $a_n\gg n^{1/(2-\delta)}$, and conversely). The Theorem and (9) are compiled on the result page theorem; the digest is on the card davenport_1963_theorem_uniform_distribution.

Covers. The corrected Statement of Problem 492 for every real sequence $a_1<a_2<\cdots$ tending to infinity with $a_{i+1}/a_i\to1$ and at most $\ll N^{2-\delta}$ terms below $N$ for some fixed $\delta>0$: for each such sequence $f(\alpha n)$ is uniformly distributed in $[0,1)$ for almost all $\alpha>0$. An infinite set of positive integers has at most $N$ terms below $N$, so the case contains every such set and answers the site's wording, with $A\subseteq\mathbb N$, yes; the problem page's Notes credit it for that. The problem as a whole is answered no on Schmidt's page, by a sequence whose gaps tend to zero; by the two theorems together, for every $\delta>0$ its number of terms below $N$ is not $O(N^{2-\delta})$ (an authored deduction).

Acceptance. Refereed: H. Davenport and P. Erdős, A theorem on uniform distribution, Magyar Tud. Akad. Mat. Kutató Int. Közl. 8 (1963), 3--11. The publication record carries no finer date than the year, so the page is named by its first day. The site's curator credits the paper, in the problem's commentary, with the case $a_n\gg n^{1/2+\epsilon}$, but the site's label DISPROVED rests on Schmidt's theorem, so the credit is not an acceptance of this case and no reviewed evidence is listed. Schmidt's 1969 introduction also attests the theorem, and no source read disputes it. The proof (pp. 5--10) is not independently reviewed here.

Depends on. Nothing on the wiki; the result rests on the refereed paper linked above.