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Claim. The Theorem and Corollary of W. M. Schmidt, Irregularities of distribution. VI, Compositio Math. 24 (1972), 63–74 (p. 64). For a sequence $\xi_1,\xi_2,\ldots$ in the unit interval let $Z(n,\alpha)$ count the $i\le n$ with $0\le\xi_i<\alpha$, let $D(n,\alpha)=\lvert Z(n,\alpha)-n\alpha\rvert$, and for $\kappa\ge0$ let $S(\kappa)$ be the set of $\alpha$ with $D(n,\alpha)\le\kappa$ for every $n$. If $d>4\kappa$, then the $d$-th derived set of $S(\kappa)$ is empty. Hence each $S(\kappa)$ is at most countable and nowhere dense, and $S(\infty)=\bigcup_\kappa S(\kappa)$, the set of anchors $\alpha$ at which $D(n,\alpha)$ stays bounded in $n$, is at most countable. So for every sequence all but countably many anchored intervals $[0,\alpha)$ have $\limsup_N\lvert D_N([0,\alpha))\rvert=\infty$, which answers Problem 255 yes and sharpens the measure-zero statement of part I of the series (Schmidt 1968), which already answers the question. The paper's closing example, a sequence for which the $d$-th derived set of $S(d)$ is nonempty for every $d$, shows that $4\kappa$ cannot be lowered to $\kappa-\varepsilon$. The source card is Schmidt 1972.

Depends on. Nothing in this wiki; the result rests on the cited paper alone.

Dating. The page is dated by the volume's year; the journal record (Compositio Math. 24 (1972), no. 1, 63–74) gives no day, and the day in the page name is a placeholder.

Acceptance. Refereed: Compositio Math. 24 (1972), no. 1, 63–74. Reviewed: the site's curator, T. F. Bloom, labels the problem PROVED (LEAN) and credits this paper in the problem's commentary with the statement that the question holds for all but countably many intervals $[0,x]$. The curator is independent of the author.