Geometry of Continued Fractions — Sail statistics

Definitions and mathematical statement

The preceding question is identified in the research summary as source code AMR-018-0017. The corresponding local record, permanent ID $1900017$, asks whether the sum of relative frequencies over all possible face types is finite in higher dimensions $n\geq3$. In schematic notation, for the intended face-type class $\mathcal F_n$ and nonnegative relative-frequency weights $w_n(F)$, its condition is \[ W_n=\sum_{F\in\mathcal F_n}w_n(F)<\infty. \] Neither the weights nor their underlying sampling measure are defined in the supplied records; they cannot simply be assumed to form an already normalized probability distribution.

The ordinary Gauss map is $T(x)=1/x-\lfloor1/x\rfloor$ on irrational $x\in(0,1)$, coding the shift of ordinary continued-fraction digits. It is explanatory background, not a definition of the requested multidimensional map.

Conditional problem. Assuming a positive answer to the preceding finiteness question, find a higher-dimensional generalization of the Gauss map for the geometric continued fractions under consideration, and compare the corresponding frequencies of sail faces with the related frequencies supplied by Möbius geometry. This task is conditional on finiteness; it does not assert that finiteness alone already constructs the map. If a probability normalization is intended, the additional condition $W_n>0$ permits weights $w_n(F)/W_n$, but the source does not state that this is the desired normalization. It specifies neither a state space and transformation nor the Möbius-geometric frequency model or comparison identity. The supplied summary reports no final comparison theorem and treats the question as unresolved, with moderate confidence.

Short English statement

If the total face-frequency mass is finite in higher dimensions, construct the appropriate Gauss-type dynamics and relate its face frequencies to those from Möbius geometry. The frequency measures and the requested relation remain unspecified.

Sources

• O. Karpenkov, Open problems in geometry of continued fractions (2017), Problem 18, https://arxiv.org/abs/1712.01450; source-TeX extraction, accessed 2026-07-29.
• Dependency explicitly named by the research summary: AMR-018-0017, the finiteness question. Its complete local record was consulted in data/unsolvedmath/problems.json, permanent ID $1900017$, corresponding to Problem 17 of the same source. It likewise leaves the relative-frequency definition implicit.
• Local registry data/unsolvedmath/problems.json, permanent ID $1900018$, source code catalogue entry, Geometry of Continued Fractions --- Sail statistics. The record labels this open and OPEN-TRIAGE, with NEEDS_REVIEW source-status and rights qualifications. No separate reference for the proposed map or Möbius comparison is supplied.