Unit Distances in Rational Directions

Definitions and mathematical statement

Let $n$ be a positive integer and let $P\subset{\mathbb{R}}^2$ be a set of $n$ distinct points in the Euclidean plane with its fixed coordinate axes. For distinct $p,q\in P$, let $\theta(p,q)\in[0,\pi)$ denote the angle of the unoriented line through $p$ and $q$, measured from the positive $x$-axis modulo $\pi$. Define the number of unordered unit-distance pairs in rational directions by \[ U_{\mathrm{rat}}(P)=\#\left\{\{p,q\}\subseteq P:\ p\ne q,\ \lVert p-q\rVert_2=1,\ \frac{\theta(p,q)}{\pi}\in{\mathbb{Q}}\right\}. \] The literal question in the supplied statement is whether \[ \forall\varepsilon>0\quad\forall n\in{\mathbb{Z}}_{\ge1}\quad\forall P\subset{\mathbb{R}}^2\text{ with }|P|=n,\qquad U_{\mathrm{rat}}(P)<n^{1+\varepsilon}. \] Here the positive-integer range for $n$ is an explicit convention; the record does not specify whether the empty configuration is included. The displayed question retains the strict inequality and coefficient $1$. No threshold $n_0(\varepsilon)$ or multiplicative constant is supplied, so an asymptotic interpretation cannot be substituted without clarification.

The supplied original statement presents this bound declaratively, whereas the cleaned statement presents it as a question. Moreover, the supplied dated literature assessment describes the extraction as a truncated theorem and conjecture fragment, reports that a rational-angle unit-distance bound is proved, and says that no complete underlying conjecture is stated for verification. Consequently, the display formalizes the literal supplied question only; the intended complete conjecture and its relation to the reported proved bound remain unresolved.

Short English statement

Count unordered pairs of distinct planar points at distance $1$ whose joining line has angle in $\pi{\mathbb{Q}}$ modulo $\pi$. The supplied question asks whether this count is strictly less than $n^{1+\varepsilon}$ for every positive integer $n$, every configuration, and every $\varepsilon>0$. The intended asymptotic qualifications and the complete underlying conjecture remain unclear.

Sources

• Local UnsolvedMath registry, permanent ID $30002466$, title Unit Distances in Rational Directions, source code catalogue entry, in the supplied Oberwolfach Reports Open Problems collection. The proposed year is recorded as 2014; no proposer is identified.
• Supplied report citation: Combinatorics (2014), Oberwolfach Reports, pp. 5--90, DOI 10.4171/owr/2014/01. Source URL and literature-source label "Original OWR record": https://doi.org/10.4171/owr/2014/01. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions; no precise location within this report is supplied for the fragment.
• Supplied literature-source label "Rational-angle unit-distance paper": https://doi.org/10.1112/S0025579311001847. No authors, title, or theorem locator are supplied for this reference.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. It labels the record open, describes the extraction as a truncated theorem and conjecture fragment, reports that the rational-angle unit-distance bound is proved, and states that no complete underlying conjecture is available for verification. The supplied background requests independent verification of time-sensitive status and cited claims. The registry also supplies the label corrected_verified and a claim of manual semantic, typographic, and LaTeX review; these are source metadata, not independent verification by this draft.