Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 255

claims/: The 3 claim pages of Problem 255, one per claimant's result; the problem's standing derives from them.

Statement. Let $z_1,z_2,\ldots \in [0,1]$ be an infinite sequence, and define the discrepancy

D_N(I) = \#\{ n\leq N : z_n\in I\} - N\lvert I\rvert.

Must there exist some interval $I\subseteq [0,1]$ such that

\limsup_{N\to \infty}\lvert D_N(I)\rvert =\infty?

Status. PROVED (LEAN): Schmidt's 1968 theorem, refereed in the Quarterly Journal of Mathematics, answers yes; see the claim page, which also links the unbuilt public Lean file that the site's Lean qualifier most likely refers to.

Source. erdosproblems.com/255, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #255, https://www.erdosproblems.com/255.

References.

[Sc68] Schmidt, Wolfgang M., Irregularities of distribution. Quart. J. Math. Oxford Ser. (2) (1968), 181-191.
[Sc72] Schmidt, Wolfgang M., Irregularities of distribution. VI. Compositio Math. (1972), 63-74.
[TiWa80] Tijdeman, R. and Wagner, G., A sequence has almost nowhere small discrepancy. Monatsh. Math. (1980), 315-329.

Formalization. None audited here. The site's Lean qualifier names no file; the 1968 claim page links the public Lean file it most likely refers to, which is not built or audited here.

Current assessment

The standing rests on three accepted claim pages, each a refereed theorem credited by the site's curator. Schmidt's 1968 theorem (Schmidt 1968) answers the question yes: for every sequence some interval has unbounded discrepancy, and the anchors $x$ at which $D_N([0,x))$ stays bounded form a set of measure zero. His 1972 countability theorem (Schmidt 1972) shows that this exceptional set is at most countable, so all but countably many anchored intervals $[0,x)$ answer the question. Tijdeman and Wagner's 1980 theorem (Tijdeman and Wagner 1980) gives the rate $\limsup_N\lvert D_N([0,x))\rvert/\log N\ge1/400$ for almost every $x$ , which the site's commentary calls essentially best possible. The site's Lean qualifier most likely refers to the public Lean file linked on the 1968 claim page, which is not built or audited here and warrants nothing. The Progress note below is author-recorded and not independently reviewed. This page records no current literature search or independent assessment of proof coverage.

Progress

The third part of Problem 1221's conjecture, $r(\mu_r-1)\to\infty$ , implies the uniform form of this problem's conclusion, that no sequence has $\sup_I|D_N(I)|$ bounded in $N$ , as the 1949 note remarks.

Known Results

Schmidt's 1968 theorem answers the question yes; his 1972 countability theorem and the Tijdeman–Wagner rate are accepted claims with their own pages, all three linked in the Current assessment.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (9)

Schmidt's theorem: some interval has unbounded discrepancy Schmidt: bounded discrepancy at only countably many anchors Tijdeman and Wagner: discrepancy of order log N at almost every anchor problems/discrepancy/E0255/claims Discrepancy discrepancy/schmidt_1972_irregularities_distribution Corollary (p. 64): bounded anchored discrepancy at only countably many points Corollary (p. 72): for Van der Corput's sequence S^(d)(d) is nonempty for every d Theorem (p. 64): the d-th derived set of S(kappa) is empty once d > 4 kappa

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