Optimal Minimal Dispersion in High Dimensions

Definitions and mathematical statement

A standard interpretation of minimal dispersion is as follows; the supplied record does not explicitly specify the class of boxes or the boundary convention, so this interpretation requires source verification. Let $n,d\in{\mathbb{Z}}_{\geq 1}$. For a set $P\subseteq[0,1]^d$ with $|P|=n$, define \[ \operatorname{disp}(P)=\sup\left\{\prod_{j=1}^{d}(b_j-a_j):\ 0\leq a_j<b_j\leq1\text{ for every }j,\quad P\cap\prod_{j=1}^{d}(a_j,b_j)=\varnothing\right\}. \] Thus the boxes in this provisional definition are open and axis-parallel, and their size is their $d$-dimensional volume. Define \[ \operatorname{disp}^{*}(n,d)=\inf_{\substack{P\subseteq[0,1]^d\\ |P|=n}}\operatorname{disp}(P). \] Under this interpretation, the explicit proposed upper-bound target is \[ \exists C>0\quad\forall d\in{\mathbb{Z}}_{\geq1}\quad\forall n\in{\mathbb{Z}}_{\geq1},\qquad n\geq d\ \Longrightarrow\ \operatorname{disp}^{*}(n,d)\leq C\frac{d}{n}, \] where $C$ is independent of $n$ and $d$. This uniform interpretation of the constant also requires confirmation from the source.

The record additionally asks for improvements to the following displayed upper and lower benchmarks: \[ \operatorname{disp}^{*}(n,d)\leq\min\left\{\frac{c_1d}{n},\frac{c_2d\log(n/d)}{n}\right\}, \qquad \operatorname{disp}^{*}(n,d)\geq\frac{\log_2 d}{4(n+\log_2 d)}. \] Here $\log_2$ denotes the base-two logarithm. The base of the unqualified $\log$, the admissible parameter range of the benchmarks, and the dependence of $c_1,c_2$ are not specified in the supplied record. The request to improve these bounds does not specify a quantitative replacement.

Unresolved inconsistency. If $c_1$ is an absolute constant, the first term of the asserted upper benchmark already gives the proposed bound $Cd/n$. Moreover, at $n=d$, the logarithmic term is zero, which is incompatible with positive minimal dispersion for a finite number of points in positive dimension under the definition above. For $n<d$, that term is negative under the usual logarithm conventions. Consequently, the upper benchmark cannot be accepted as a valid bound over all positive $n,d$, or over the entire stated target range $n\geq d$. The intended formula, constant dependence, or restricted range must be checked; no correction is inferred here.

The supplied registry labels this problem partially_solved. Its supplied dated assessment says that subsequent work improves both upper and lower bounds and obtains near-matching high-dimensional estimates, while the conjectured order without the remaining logarithmic losses is not proved in full generality. This is supplied curation, not an independently verified mathematical or literature assessment.

Short English statement

Under a provisional standard definition, minimal dispersion is the smallest possible supremum of the volumes of empty axis-parallel boxes among configurations of $n$ points in $[0,1]^d$. The intended question seeks a uniform bound $Cd/n$ for $n\geq d$ and improved lower bounds. The supplied upper benchmark is internally inconsistent with that target and needs source verification before a definitive formulation is possible.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30003395$, title Optimal Minimal Dispersion in High Dimensions, source problem code catalogue entry; supplied collection: Oberwolfach Reports Open Problems. No individual proposer is supplied.
• Supplied report citation: Mini-Workshop: Perspectives in High-Dimensional Probability and Convexity (2018), Oberwolfach Reports, pp. 499--526, DOI 10.4171/owr/2017/10, https://doi.org/10.4171/owr/2017/10. The record gives proposed year 2017, whereas the citation labels the workshop 2018; this discrepancy is unresolved.
• Supplied literature source, "Oberwolfach report": https://ems.press/content/serial-article-files/46672.
• Supplied literature source, "Arman--Litvak": https://doi.org/10.1016/j.jco.2024.101883.
• Supplied literature source, "Trödler--Volec--Vybíral": https://arxiv.org/abs/2311.10666.
• Literature assessment checked at 2026-08-21, according to the supplied record: status partially_solved; later work improves both upper and lower bounds and obtains near-matching high-dimensional estimates, but the conjectured order without the remaining logarithmic losses is not proved in full generality. This assessment is supplied dated curation derived from the OWR collection, not independently verified here; the time-sensitive status and cited claims require independent verification.