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Claim. There is a strictly increasing real sequence $x_0=0<x_1<x_2<\cdots$ with $x_n\to\infty$ and $x_{n+1}/x_n\to1$ such that the multiples $\alpha,2\alpha,\ldots$ are not uniformly distributed relative to $(x_n)$ for almost every $\alpha>0$. Schmidt's setting (p. 137): for $0<\lambda\le1$, $M(\lambda)$ is the union of the lower $\lambda$-parts $[x_n,x_n+\lambda(x_{n+1}-x_n))$ of the gaps, $n\ge0$; the multiples are uniformly distributed relative to $(x_n)$ when the proportion of $k\le N$ with $k\alpha\in M(\lambda)$ tends to $\lambda$ for every $\lambda$; and $f(x)$ is $1$ on $M(1/2)$ and $-1$ elsewhere. The theorem is printed as "Theorem 1. There is a function $f(x)$ of the type considered above such that (6) $\limsup_{N\to\infty}|N^{-1}\sum_{n=1}^Nf(\alpha n)|=1$ for almost every $\alpha>0$" (p. 137). In the problem page's notation, with $a_i=x_i$ for $i\ge1$ (Schmidt's $x_0=0$ lies below $a_1$), the position $f(y)$ of $y$ within its gap is below $1/2$ exactly when $y\in M(1/2)$, so Schmidt's test function is $2\cdot\mathbf 1[f<1/2]-1$ and its averages are $2N^{-1}\#\{n\le N:f(\alpha n)<1/2\}-1$; uniform distribution of the positions would force these to tend to $0$, and (6) says their absolute value returns to $1$ for almost every $\alpha>0$ (an authored translation, recorded on the problem page). The finitely many $n$ with $\alpha n<a_1$, where the problem's $f$ is undefined and Schmidt's first interval starts at $x_0=0$, do not affect the limit. So the sequence $a_1<a_2<\cdots$ tends to infinity with $a_{i+1}/a_i\to1$, and for almost every $\alpha>0$ the sequence $f(\alpha n)$ is not uniformly distributed: the corrected Statement of [[problems/number_theory/E0492/_index|Problem 492]] is false. The theorem is compiled on the result page theorem_1; the digest is on the card schmidt_1969_disproof_conjectures_diophantine_approximations.

Argument, in outline. Lemma 1 (pp. 138--140) builds, for given $N$ and $\varepsilon$, a subdivision of the unit interval with mesh below $\varepsilon$ whose test function satisfies $f(x)=f(mx)$ for every integer $m$ with $1\le m\le N$ whenever $x$ and $mx$ lie in the unit interval, outside a set of measure below $\varepsilon$, by Dirichlet's simultaneous approximation of the numbers $\log m$; Section 3 (pp. 140--141) glues scaled copies on blocks $[N_{k-1},N_k)$ with $N_k\ge2k^2N_{k-1}$, so that the $k$-th block has gaps $x_{n+1}-x_n\le1/k$ and, for $\alpha$ in a fixed interval outside exceptional sets of measure at most $\varepsilon_kN_k<1/k$, which tends to zero, so that almost every $\alpha$ avoids infinitely many of them, the values $f(m\alpha)$ agree for $N_k/k\le m<N_k/b$, which gives (6). The proof is not independently checked here.

Formalization. Collin Yuanjie Ren's AI-assisted Lean development, linked above at a pinned commit, formalizes Schmidt's counterexample in the form that the multiples are not uniformly distributed relative to the constructed sequence for almost every $\alpha>0$, not the exact limsup (6); its root is Erdos492Real.schmidt_counterexample. It is third-party Lean, not built here, so no formalized evidence is listed.

Acceptance. Refereed: W. M. Schmidt, Disproof of some conjectures on Diophantine approximations, Studia Sci. Math. Hungar. 4 (1969), 137--144, received 2 April 1968 (the paper's running header misprints the volume as "3 (1968)"); the record carries no finer publication date than the year, so the page is named by its first day. Reviewed: the site's curator, Thomas F. Bloom, labels the problem DISPROVED and credits Schmidt, in the problem's commentary, with showing that the general conjecture is false; the curator neither wrote nor submitted the result. Nothing here is independently reviewed by this project.

Depends on. Nothing on the wiki; the construction is self-contained in the refereed paper linked above.