Discrepancy Lower Bounds for Line Unions in Three Dimensions

Definitions and mathematical statement

Let $N\geq 1$ be an integer and let $\ell_1,\ldots,\ell_N\subset{\mathbb{R}}^3$ be affine lines. Set \[ A=\bigcup_{i=1}^{N}(\ell_i\cap[0,1]^3),\qquad L=H^1(A), \] where $H^1$ denotes length and $H^3$ denotes three-dimensional volume, using the usual normalization of Hausdorff measure. The measure is that of the union as a set, without multiplicity.

The supplied statement does not specify the family of test sets over which $Q$ ranges. Denote that intended, but unresolved, family by $\mathcal Q$. For a specified nonempty family of measurable test sets and for $L>0$, the displayed discrepancy would be \[ D_N(A)=\sup_{Q\in\mathcal Q}\left|\frac{H^1(A\cap Q)}{L}-H^3(Q)\right|. \] The proposed question is whether there exists an absolute constant $c>0$, independent of $N$ and of the lines, such that for every such union with positive length, \[ D_N(A)\geq c\,\frac{\log L}{L}, \] where $\log$ is the natural logarithm.

This is a conditional formulation: the intended family $\mathcal Q$, including its shapes, location, anchoring, and boundary conventions, must be recovered before the question is fully specified. The supplied universal wording also permits unions of zero length, for which both displayed quotients are undefined. The restriction $L>0$ is necessary to interpret these formulas but is not explicitly supplied. For $0<L\leq 1$, the proposed lower bound is automatically satisfied because its right-hand side is nonpositive; its substantive range is $L>1$. No further length threshold is supplied.

Short English statement

The question asks whether the discrepancy of normalized length on every finite union of lines clipped to the unit cube has a universal lower bound of order $\log L/L$, where $L$ is the total length of the union. The test-set family is unspecified, and zero-length unions make the formulas undefined, so these qualifications require clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002421$, source problem code catalogue entry, titled Discrepancy Lower Bounds for Line Unions in Three Dimensions, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2013 and names no proposer.
• Supplied report citation: Uniform Distribution Theory and Applications (2014), Oberwolfach Reports, pp. 2837--2917, DOI 10.4171/owr/2013/49. Source URL and supplied literature reference labeled "original OWR report": https://doi.org/10.4171/owr/2013/49. The supplied background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature assessment, checked on 2026-08-21: the record labels the problem open and states that the proposed bound would extend Schmidt-type bounds beyond point sets, while the checked primary literature gives no proof or disproof. This is supplied dated curation, not an independently verified literature assessment. The record also labels the statement corrected_verified and reports comparison with the official report, with notation, context, and question form normalized without changing the source-level claim; that verification claim has not been independently checked here.