Geometry of Continued Fractions — Sail statistics

Definitions and mathematical statement

For each dimension parameter $n$, let $\mathcal F_n$ denote the intended set of possible face types in the sail-statistics setting, and let $r_n(F)\ge0$ denote the intended relative frequency of $F\in\mathcal F_n$. The source asks whether \[ \forall n\ge3,\qquad \sum_{F\in\mathcal F_n}r_n(F)<\infty. \] For a nonnegative family, the sum can be defined without an ordering by \[ \sum_{F\in\mathcal F_n}r_n(F) =\sup_{\mathcal A\subseteq\mathcal F_n\text{ finite}} \sum_{F\in\mathcal A}r_n(F). \] Thus the requested finiteness means a finite bound for this sum in each fixed dimension, not a bound uniform over all dimensions.

The record does not define the relative frequencies, their sampling measure or limiting process, the normalization, the equivalence relation on faces, or which face dimensions are included. It also does not specify whether $n$ is ambient dimension or sail dimension. The displayed statement is therefore conditional on those source definitions, which cannot be reconstructed uniquely from the excerpt. In particular, interpreting the frequencies as a probability distribution on disjoint exhaustive face categories would force a normalized sum and need not express the intended problem.

The supplied moderate-confidence OPEN-TRIAGE summary reports finiteness in dimension $2$, relating it to Gauss-map ergodicity and Karpenkov/Arnold statistics, and says finiteness for $n\ge3$ was unresolved in the located literature. It relates the question to source items AMR-018-0016 and AMR-018-0018. These are supplied status and contextual claims, not a new literature assessment.

Short English statement

In each higher dimension, is the total of the relative frequencies of all sail face types finite? The definition and normalization of frequency and the class of face types are missing from the record.

Sources

• Local UnsolvedMath registry, permanent ID $1900017$, exact source code catalogue entry, "Geometry of Continued Fractions --- Sail statistics."
• Karpenkov, Open problems in geometry of continued fractions (2017), Problem 17; https://arxiv.org/abs/1712.01450. Source-TeX extraction; recorded access 2026-07-29.
• Supplied research summary mentions Karpenkov's frequency papers, Arnold's statistics, Gauss-map ergodicity, and related source items AMR-018-0016/AMR-018-0018; it supplies no complete additional references or frequency definitions.