Farey Sequences for Thin Groups

Christopher Lutsko

arxiv: 1907.01854 · v1 · pith:XQHFJS3Vnew · submitted 2019-07-03 · 🧮 math.DS · math.NT

Farey Sequences for Thin Groups

Christopher Lutsko This is my paper

Pith reviewed 2026-05-25 09:50 UTC · model grok-4.3

classification 🧮 math.DS math.NT

keywords Farey sequencesthin groupsequidistributiongap distributionFuchsian groupsGauss measureDiophantine approximationcontinued fractions

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The pith

Generalized Farey sequences from thin discrete groups equidistribute and their gaps converge in distribution.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines generalized Farey sequences using the action of thin discrete subgroups of the modular group. It proves that these sequences become equidistributed in the unit interval as the height grows. The normalized gaps between consecutive terms converge in distribution to a limiting law. For sequences with continued fraction partial quotients satisfying congruence conditions, this gives new results in Diophantine approximation. In a concrete example the author constructs an explicit gap distribution and an ergodic invariant measure analogous to the Gauss measure.

Core claim

Generalized Farey sequences arise from orbits under thin Fuchsian groups and equidistribute with respect to Lebesgue measure. Their gap distributions converge, and for one example an explicit formula is derived from a sparse Ford configuration. The associated Gauss-like measure is shown to be ergodic under the Gauss map, which implies Gauss-Kuzmin statistics for the sequence.

What carries the argument

The generalized Farey sequence obtained from the orbit of infinity under a thin subgroup, together with the associated Ford circles and the induced Gauss map on the interval.

If this is right

The sequences equidistribute in the unit interval.
The gap distribution converges to a limiting distribution.
An associated Diophantine approximation problem for Fuchsian groups is solved.
For a specific example, an explicit formula for the gap distribution is obtained via a sparse Ford configuration.
The analogue of the Gauss measure is ergodic for the Gauss map and yields Gauss-Kuzmin statistics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This construction could apply to other arithmetic constraints on continued fraction expansions beyond simple congruences.
The ergodicity might imply stronger mixing properties for the associated dynamical system.
Similar gap and equidistribution results may hold for thin groups acting in higher dimensions or on other hyperbolic spaces.

Load-bearing premise

The equidistribution and gap techniques from the classical case extend to these sequences generated by thin groups without needing additional assumptions on the group structure.

What would settle it

Computing the empirical gap distribution for the specific example and finding it does not match the explicit formula derived from the sparse Ford configuration would falsify the convergence claim.

read the original abstract

The classical Farey sequence of height $Q$ is the set of rational numbers in reduced form with denominator less than $Q$. In this paper we introduce the concept of a generalized Farey sequence. While these sequences arise naturally in the study of discrete (and in particular thin) subgroups, they can be used to study interesting number theoretic sequences - for example rationals whose continued fraction partial quotients are subject to congruence conditions. We show that these sequences equidistribute, that the gap distribution converges, and we answer an associated problem in Diophantine approximation with Fuchsian groups. Moreover, for one specific example, we use a sparse Ford configuration construction to write down an explicit formula for the gap distribution. Finally for this example, we construct the analogue of the Gauss measure in this context which we show is ergodic for the Gauss map. This allows us to prove a theorem about the Gauss-Kuzmin statistics of the sequence.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Lutsko defines generalized Farey sequences from thin Fuchsian groups, proves equidistribution and gap convergence in general, and gives an explicit gap formula plus ergodic Gauss analogue for one concrete case.

read the letter

The core contribution is the construction of these generalized Farey sequences tied to thin discrete subgroups and the proof that they equidistribute with convergent gap distributions. He also settles a related Diophantine approximation question for Fuchsian groups and, in one explicit example, produces a closed-form gap distribution via sparse Ford configurations plus an ergodic invariant measure for the Gauss map that yields Gauss-Kuzmin statistics. That example is the clearest part of the work and shows the ideas can be made concrete. The framing around congruence-constrained continued fractions is a reasonable motivation and the results look like a direct extension of classical techniques rather than a routine one. The general theorems are stated cleanly in the abstract. The main uncertainty is whether the equidistribution and gap arguments carry over without extra hypotheses. Thin groups have limit sets of Lebesgue measure zero, so any step that leans on positive-measure invariant measures or mixing on the full circle needs careful justification; the abstract does not list compensating conditions such as geometric finiteness or critical-exponent bounds. The specific example sidesteps this by construction, but the general case will stand or fall on the details of the proofs. This is aimed at researchers already working in thin groups, hyperbolic dynamics, and constrained Diophantine approximation. Someone in that niche will find the constructions and the worked example useful. The paper is specific enough and the claims are grounded enough to merit referee time rather than a desk rejection.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces generalized Farey sequences arising naturally from the action of thin discrete subgroups of Fuchsian groups. It proves that these sequences equidistribute in the unit interval, that their gap distributions converge, and resolves an associated Diophantine approximation problem. For one specific example it constructs a sparse Ford configuration yielding an explicit gap distribution formula, defines an analogue of the Gauss measure, proves its ergodicity under the Gauss map, and deduces Gauss-Kuzmin statistics for the sequence.

Significance. If the central claims hold, the work extends classical equidistribution and gap-distribution results for Farey sequences to the thin-group setting (limit sets of Lebesgue measure zero), which is a substantive advance for Diophantine approximation and hyperbolic dynamics. The explicit sparse-Ford construction and the ergodicity proof for the specific example constitute concrete, verifiable contributions that can be checked independently.

major comments (1)

[Abstract and §2] Abstract, paragraph 2 and §2 (definition of generalized Farey sequences): the assertion that classical equidistribution and gap techniques extend to arbitrary thin groups without additional structural hypotheses is load-bearing for the general theorems, yet the zero Lebesgue measure of the limit set for thin groups (critical exponent <1) raises the question whether the proofs rely on invariant measures supported on a positive-measure set or on mixing rates that fail to carry over; the manuscript must state explicitly the compensating conditions (e.g., geometric finiteness, lower bound on the critical exponent, or a uniform sparse-Ford property) that make the extension valid.

minor comments (2)

[§1] The notation for the generalized Farey sequence of height Q should be introduced with a displayed definition and at least one concrete numerical example before the statements of the main theorems.
[Figures 1–3] Figure captions for the Ford configurations should include the precise group and the value of the critical exponent used in each panel.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed report and the opportunity to clarify the hypotheses underlying our results. We address the major comment below and will revise the manuscript to make the required conditions explicit.

read point-by-point responses

Referee: [Abstract and §2] Abstract, paragraph 2 and §2 (definition of generalized Farey sequences): the assertion that classical equidistribution and gap techniques extend to arbitrary thin groups without additional structural hypotheses is load-bearing for the general theorems, yet the zero Lebesgue measure of the limit set for thin groups (critical exponent <1) raises the question whether the proofs rely on invariant measures supported on a positive-measure set or on mixing rates that fail to carry over; the manuscript must state explicitly the compensating conditions (e.g., geometric finiteness, lower bound on the critical exponent, or a uniform sparse-Ford property) that make the extension valid.

Authors: The proofs of equidistribution and convergence of gap distributions rely on the existence of a Patterson-Sullivan measure for the group action together with a spectral gap coming from the critical exponent δ > 1/2; these properties hold for geometrically finite thin subgroups of PSL(2,R) and do not require positive Lebesgue measure on the limit set. The manuscript implicitly uses geometric finiteness throughout (as is standard for thin groups admitting a Ford fundamental domain), but does not spell out the lower bound on δ or the geometric-finiteness assumption in §2. We will add an explicit paragraph in the introduction and at the beginning of §2 listing the standing hypotheses on the groups (geometric finiteness and δ > 1/2) under which the classical techniques carry over, together with a brief justification why zero Lebesgue measure is not an obstruction. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations are independent of target statistics.

full rationale

The abstract and description present generalized Farey sequences arising from thin groups, with claims of equidistribution, gap convergence, an explicit gap formula via sparse Ford configuration for one example, and ergodicity of an analogue Gauss measure. No quoted equations, definitions, or steps show any result reducing by construction to a fitted parameter, self-definition, or self-citation chain that forces the outcome. The constructions are presented as extensions of classical methods, remaining self-contained against external benchmarks without the target statistics being presupposed in the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper is a pure-mathematics work in dynamical systems and number theory. No free parameters or invented entities are visible from the abstract. It relies on standard background results about discrete subgroups, continued fractions, and ergodic theory.

axioms (2)

standard math Standard properties of discrete subgroups of SL(2,R) and the geometry of Ford circles
Invoked when defining the generalized sequences and the sparse Ford configuration.
standard math Existence and basic ergodicity properties of the classical Gauss measure
Used as the model for the analogue measure constructed in the example.

pith-pipeline@v0.9.0 · 5676 in / 1461 out tokens · 46331 ms · 2026-05-25T09:50:21.006562+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We show that these sequences equidistribute, that the gap distribution converges... construct the analogue of the Gauss measure... ergodic for the Gauss map (abstract).
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Γ < PSL(2,R) ... critical exponent δ_Γ ... Patterson-Sullivan measure μ_PS (Section 2).

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