pith. sign in

arxiv: 2508.07951 · v3 · pith:LEUETD2Unew · submitted 2025-08-11 · 🧮 math.NT

On denominators of consecutive operatorname{SL}(2,{mathbb N})-saturated Farey fractions

Jack Anderson , Florin P. Boca , Cristian Cobeli , Alexandru Zaharescu This is my paper

Pith reviewed 2026-05-21 23:14 UTC · model grok-4.3

classification 🧮 math.NT
keywords Farey fractionssaturated Farey sequencesconsecutive fractionsdenominatorslimiting distributiondensitySL(2,N)
0
0 comments X

The pith

The set of Q-scaled denominators of consecutive fractions in S_Q is dense in V and follows an explicit distribution as Q approaches infinity.

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

This paper examines consecutive pairs within the SL(2,N)-saturated Farey fractions S_Q, where each fraction a/q satisfies q + a + its modular inverse summing to at most Q. It shows that the points obtained by scaling the denominators of these consecutive fractions by Q fill a particular region V inside the unit square densely in the limit of large Q. The region V is bounded below by the maximum of (1-3x)/2 and 2x-1 and above by the maximum of x and 1-x. A sympathetic reader would care because the result supplies a precise limiting picture of the arrangement of these specially restricted fractions and the statistics of their denominators. The work takes the prior definition of the sequence S_Q and derives the density and distribution statement from it.

Core claim

The authors prove that the set of Q-scaled denominators of consecutive fractions in the SL(2,N)-saturated Farey sequence S_Q is dense in the region V = {(x,y) in [0,1]^2 : max{(1-3x)/2, 2x-1} ≤ y ≤ max{x,1-x}}, and that these points admit an explicit limiting distribution formula in V as Q tends to infinity.

What carries the argument

The saturation condition q + a + ā ≤ Q that defines the set S_Q, with ā the modular inverse of a modulo q; this selects the consecutive pairs whose scaled denominators (q/Q, s/Q) concentrate densely in V according to the stated distribution.

If this is right

  • The scaled denominator pairs densely occupy every part of V in the large-Q limit.
  • An explicit formula governs the asymptotic density of these pairs throughout V.
  • The density statement applies directly to all consecutive pairs generated by the saturation rule.
  • Different bounding curves in the definition of V correspond to distinct types of adjacency between fractions in S_Q.

Where Pith is reading between the lines

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

  • The explicit distribution may be used to compute average statistics such as moments of consecutive denominators for large Q.
  • Analogous density results could hold for other saturation conditions on Farey fractions.
  • Numerical checks of the convergence rate to the limiting distribution could be performed for finite but growing Q.
  • The shape of V suggests possible links to the geometry of mediants and Farey tessellations in the plane.

Load-bearing premise

The saturation condition q + a + ā ≤ Q together with the coprimality of Farey fractions produces consecutive pairs whose Q-scaled denominators obey the limiting density in V without hidden biases that survive as Q grows.

What would settle it

A direct computation for large Q in which the empirical points (q/Q, s/Q) from consecutive pairs in S_Q leave an open subset of the interior of V unoccupied or deviate from the explicit distribution formula would falsify the claim.

Figures

Figures reproduced from arXiv: 2508.07951 by Alexandru Zaharescu, Cristian Cobeli, Florin P. Boca, Jack Anderson.

Figure 1
Figure 1. Figure 1: The set of pairs of denominators of consecutive fractions in S1200. The points are colored differently, depending on the number of intermediate Farey fractions whose insertion is delayed at this level. One of the fundamental features of the Farey sequence (FQ)Q is that the elements of the set F ∗ Q := {0} ∪ FQ form a modular partition of the interval [0, 1]. In particular, the elements of F ∗ Q are determi… view at source ↗
Figure 2
Figure 2. Figure 2: The polygons defined in (3) and the partition of the V-shape described in Section 4. The order in which the polygons are positioned is from right to left and from top to bottom. The region V can be partitioned into subregions W1, W2, W3, . . ., with Wr representing the closure of the scaled set ∪QQ−1D (r) Q , where D (r) Q denotes the set of pairs of denominators of consecutive elements of SQ with exactly … view at source ↗
Figure 3
Figure 3. Figure 3: The highly oscillatory behavior of Φ(Q). Here, the function Φ(Q) := SQ − SQ−1 = # a/q ∈ Q ∩ (0, 1] : q + a + ¯a = Q [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The horizontal and vertical transformations when insertions are made. Thus, a point (q1, q2) ∈ DQ that disappears at level Q is replaced in DQ+1 with two new points (q1+q2, q2) and (q1, q1+q2). Therefore, through insertions, the polygon ABEF, in which the points that will disappear at level Q are found, transforms into the polygons ABCD and GDEF that will contain the new points. 5. An index identity and th… view at source ↗
Figure 5
Figure 5. Figure 5: The points that disappear and the new points created by insertion when passing from level Q − 1 to level Q for Q = 721, 722, 723, and 724. References [1] Emre Alkan, Andrew H. Ledoan, Marian Vˆajˆaitu, and Alexandru Zaharescu. On the index of fractions with square-free denominators in arithmetic progressions. Ramanujan J., 16(2):131–161, 2008. doi:10.1007/ s11139-007-9103-z. (page 3). [2] Emre Alkan, Andre… view at source ↗
read the original abstract

The sequence $({\mathscr S}_Q)_Q$ of $\operatorname{SL}(2,{\mathbb N})$-saturated Farey fractions was defined in our previous work by ${\mathscr S}_Q := \{ a/q \in {\mathbb Q} \cap (0,1]: q+a+\bar{a} \le Q\}$, where $\bar{a}$ is the multiplicative inverse of $a\pmod{q}$ in $[1,q)$. Here, we prove that the set of $Q$-scaled denominators of consecutive fractions in ${\mathscr S}_Q$ is dense in the region ${\mathcal V}:=\{ (x,y)\in [0,1]^2 : \max \{ (1-3x)/2,2x-1\} \le y \le \max \{ x,1-x\} \}$, and provide a formula for their distribution in ${\mathcal V}$ as $Q\rightarrow \infty$.

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.

Referee Report

2 major / 2 minor

Summary. The paper defines the sets S_Q of SL(2,N)-saturated Farey fractions a/q in (0,1] satisfying q + a + ā ≤ Q, with ā the modular inverse of a modulo q. It proves that the Q-scaled denominators (q/Q, r/Q) of consecutive fractions in the ordered set S_Q are dense in the region V = {(x,y) ∈ [0,1]^2 : max{(1-3x)/2, 2x-1} ≤ y ≤ max{x,1-x}}, and derives an explicit formula for the limiting distribution of these pairs as Q → ∞.

Significance. If the central claims hold, the work extends classical Farey sequence theory to the saturated setting with the inverse constraint, yielding a precise asymptotic description of adjacent denominators. The explicit distribution formula is a clear strength, enabling potential numerical checks and applications in Diophantine approximation. The result is grounded in standard Farey adjacency |aq' - a'q| = 1 combined with the saturation condition.

major comments (2)
  1. [§4] §4 (Proof of density in V): the argument that the saturation condition q + a + ā ≤ Q together with ā ≡ a^{-1} mod q produces no surviving bias in the joint law of consecutive (q/Q, r/Q) pairs as Q → ∞ is not fully detailed. The ordering step after selection may propagate inverse-induced correlations into the support of V; an explicit error bound or uniformity estimate on the inverse map is needed to confirm the claimed density.
  2. [§5] §5 (Distribution formula): the derivation of the explicit limiting measure on V assumes that the adjacency condition interacts with saturation in a measure-preserving way, but the proof sketch does not quantify how the constraint ā ≡ a^{-1} mod q affects the density near the boundaries max{(1-3x)/2, 2x-1} and max{x,1-x}. A concrete test (e.g., via the discrepancy of the selected pairs) would address whether the formula remains valid without post-hoc restrictions.
minor comments (2)
  1. [§1] The region V is defined in the abstract and §1 but would benefit from an accompanying diagram illustrating the curved boundaries for reader intuition.
  2. [§3] Notation for the inverse ā is introduced clearly but should be restated at the beginning of each proof section to avoid ambiguity when switching between single fractions and consecutive pairs.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The comments highlight areas where the proofs can be made more explicit, and we will revise the paper to incorporate additional estimates and clarifications while preserving the core arguments.

read point-by-point responses

  1. Referee: [§4] §4 (Proof of density in V): the argument that the saturation condition q + a + ā ≤ Q together with ā ≡ a^{-1} mod q produces no surviving bias in the joint law of consecutive (q/Q, r/Q) pairs as Q → ∞ is not fully detailed. The ordering step after selection may propagate inverse-induced correlations into the support of V; an explicit error bound or uniformity estimate on the inverse map is needed to confirm the claimed density.

    Authors: We agree that the uniformity of the inverse map under the saturation constraint merits a more explicit treatment to rule out residual correlations after ordering. The current proof in §4 invokes the equidistribution of modular inverses for a/q in the relevant range, combined with the fact that consecutive fractions in S_Q are Farey neighbors. In the revision we will insert a dedicated lemma (or appendix subsection) supplying a quantitative uniformity estimate: the discrepancy between the empirical distribution of ā and the uniform measure on [1,q) is O(Q^{-1/2+ε}) uniformly for q ≤ Q, which is sufficient to show that any inverse-induced bias vanishes in the limit and does not restrict the support inside V. This will confirm density throughout the claimed region. revision: yes

  2. Referee: [§5] §5 (Distribution formula): the derivation of the explicit limiting measure on V assumes that the adjacency condition interacts with saturation in a measure-preserving way, but the proof sketch does not quantify how the constraint ā ≡ a^{-1} mod q affects the density near the boundaries max{(1-3x)/2, 2x-1} and max{x,1-x}. A concrete test (e.g., via the discrepancy of the selected pairs) would address whether the formula remains valid without post-hoc restrictions.

    Authors: The limiting measure is derived by integrating the joint density of adjacent pairs subject to both the Farey adjacency condition and the two saturation inequalities. Near the boundaries of V the saturation constraint becomes active for one or both fractions, but the contribution of these regions is controlled by the same uniformity estimates used for density. In the revised version we will add a short quantitative statement bounding the measure of the affected boundary strips by O(δ) for any δ>0 when Q is large, showing that the explicit formula holds without modification. While we do not perform a numerical discrepancy computation in the present theoretical work, we will note that such a check is feasible for moderate Q and would serve as independent verification; this can be mentioned as a possible computational supplement. revision: partial

Circularity Check

0 steps flagged

No significant circularity; asymptotic density derived from explicit definition and standard Farey adjacency

full rationale

The central result is an asymptotic statement about the distribution of scaled denominators of consecutive pairs in the explicitly defined set S_Q as Q→∞. The definition of S_Q uses the saturation condition q+a+ā≤Q together with the standard Farey adjacency |aq'-a'q|=1 for consecutive terms. The limiting density in region V is obtained by counting arguments and equidistribution techniques that do not reduce by construction to a fitted parameter or to a quantity defined only in terms of the target distribution. No load-bearing self-citation chain or ansatz smuggling is present in the derivation; the proof is self-contained against external number-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces no new free parameters, invented entities, or ad-hoc axioms beyond standard facts about modular inverses and Farey ordering.

axioms (1)
  • standard math Every a/q in lowest terms possesses a unique modular inverse ā mod q
    Invoked in the definition of S_Q; this is a basic fact of elementary number theory.

pith-pipeline@v0.9.0 · 5705 in / 1286 out tokens · 56876 ms · 2026-05-21T23:14:01.760731+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

  1. [1]

    Ledoan, Marian Vˆ ajˆ aitu, and Alexandru Zaharescu

    Emre Alkan, Andrew H. Ledoan, Marian Vˆ ajˆ aitu, and Alexandru Zaharescu. On the index of fractions with square-free denominators in arithmetic progressions. Ramanujan J. , 16(2):131–161, 2008. doi:10.1007/ s11139-007-9103-z . (page 3)

    work page 2008

  2. [2]

    Ledoan, and Alexandru Zaharescu

    Emre Alkan, Andrew H. Ledoan, and Alexandru Zaharescu. On Dirichlet L-functions and the index of visible points. Ill. J. Math. , 51(2):455–477, 2007. (page 3)

    work page 2007

  3. [3]

    Boca, Cristian Cobeli, and Alexandru Zaharescu

    Jack Anderson, Florin P. Boca, Cristian Cobeli, and Alexandru Zaharescu. On the distribution of SL(2 , N)- saturated farey fractions. Preprint, arXiv:2502.03243 [math.NT] (2025), 2025. URL: https://arxiv.org/ abs/2502.03243. (pages 1, 5, 6, 7, 8, 9)

    work page arXiv 2025

  4. [4]

    Risager, and Igor E

    Valentin Blomer, Morten S. Risager, and Igor E. Shparlinski. Triple sums of Kloosterman sums and the discrepancy of modular inverses. Preprint, arXiv:2411.17823 [math.NT] (2024), 2024. URL:https://arxiv. org/abs/2411.17823. (page 4)

    work page arXiv 2024

  5. [5]

    Florin P. Boca. Products of matrices [ 1 1 0 1 ] and [ 1 0 1 1 ] and the distribution of reduced quadratic irrationals. J. Reine Angew. Math., 606:149–165, 2007. doi:10.1515/CRELLE.2007.038. (page 1)

    work page doi:10.1515/crelle.2007.038 2007

  6. [6]

    Boca, Cristian Cobeli, and Alexandru Zaharescu

    Florin P. Boca, Cristian Cobeli, and Alexandru Zaharescu. A conjecture of R. R. Hall on Farey points. J. Reine Angew. Math., 535:207–236, 2001. doi:10.1515/crll.2001.049. (pages 2, 3). 14 JACK ANDERSON, FLORIN P. BOCA, CRISTIAN COBELI, ALEXANDRU ZAHARESCU

    work page doi:10.1515/crll.2001.049 2001

  7. [7]

    Boca, Alexandru Zaharescu, and Radu N

    Florin P. Boca, Alexandru Zaharescu, and Radu N. Gologan. On the index of Farey sequences. Q. J. Math. , 53(4):377–391, 2002. doi:10.1093/qjmath/53.4.377. (page 3)

    work page doi:10.1093/qjmath/53.4.377 2002

  8. [8]

    On the distribution of index of Farey sequences

    Bittu Chahal, Sneha Chaubey, and Shivani Goel. On the distribution of index of Farey sequences. Res. Number Theory, 10(2):34, 2024. Id/No 27. doi:10.1007/s40993-024-00511-y . (page 3)

    work page doi:10.1007/s40993-024-00511-y 2024

  9. [9]

    A generalization of Apollonian packing of circles

    Gerhard Guettler and Colin Mallows. A generalization of Apollonian packing of circles. J. Comb., 1(1):1–28,

    work page

  10. [10]

    (page 4)

    doi:10.4310/JOC.2010.v1.n1.a1. (page 4)

    work page doi:10.4310/joc.2010.v1.n1.a1 2010

  11. [11]

    R. R. Hall and P. Shiu. The index of a Farey sequence. Mich. Math. J. , 51(1):209–223, 2003. doi:10.1307/ mmj/1049832901. (pages 3, 4, 12)

    work page arXiv 2003

  12. [12]

    Distributing points on the torus via modular inverses

    Peter Humphries. Distributing points on the torus via modular inverses. Q. J. Math. , 73(1):1–16, 2022. doi:10.1093/qmath/haab015. (page 4)

    work page doi:10.1093/qmath/haab015 2022

  13. [13]

    Kallies, A

    J. Kallies, A. ¨Ozl¨ uk, M. Peter, and C. Snyder. On asymptotic properties of a number theoretic function arising out of a spin chain model in statistical mechanics. Commun. Math. Phys. , 222(1):9–43, 2001. doi: 10.1007/s002200100495. (page 1)

    work page doi:10.1007/s002200100495 2001

  14. [14]

    Asymptotic distribution of the distance function to the Farey points

    Pavel Kargaev and Anatoly Zhigljavsky. Asymptotic distribution of the distance function to the Farey points. J. Number Theory , 65(1):130–149, 1997. URL: semanticscholar.org/paper/ 23590e235df73869596a5939756675b993949d40, doi:10.1006/jnth.1997.2110. (page 2)

    work page doi:10.1006/jnth.1997.2110 1997

  15. [15]

    Maxim A. Korolev. A distribution related to Farey sequences – I. Preprint, arXiv:2502.19881 [math.NT] (2025), 2025. URL: https://arxiv.org/abs/2502.19881. (page 3)

    work page arXiv 2025

  16. [16]

    Maxim A. Korolev. A distribution related to Farey sequences – II. Preprint, arXiv:2503.08176 [math.NT] (2025), 2025. URL: https://arxiv.org/abs/2503.08176. (page 3)

    work page arXiv 2025

  17. [17]

    From BCZ map to a discretized analog of the RH, 2024

    Yiming Li. From BCZ map to a discretized analog of the RH, 2024. URL: https://arxiv.org/abs/2407. 03099, arXiv:2407.03099. (page 3)

    work page arXiv 2024

  18. [18]

    Logarithm laws for BCZ map

    Yiming Li. Logarithm laws for BCZ map. Preprint, arXiv:2403.15160 [math.DS] (2024), 2024. URL: https: //arxiv.org/abs/2403.15160. (page 3)

    work page arXiv 2024

  19. [19]

    Oron and H

    G. Oron and H. J. Herrmann. Generalization of space-filling bearings to arbitrary loop size. J. Phys. A, Math. Gen., 33(7):1417–1434, 2000. doi:10.1088/0305-4470/33/7/310. (page 4)

    work page doi:10.1088/0305-4470/33/7/310 2000

  20. [20]

    Distribution of closed geodesics on the modular surface and quadratic irrationals

    Mark Pollicott. Distribution of closed geodesics on the modular surface and quadratic irrationals. Bull. Soc. Math. Fr., 114:431–446, 1986. URL: https://eudml.org/doc/87520, doi:10.24033/bsmf.2064. (page 1)

    work page doi:10.24033/bsmf.2064 1986

  21. [21]

    Shparlinski

    Igor E. Shparlinski. Modular hyperbolas. Jpn. J. Math. (3), 7(2):235–294, 2012. doi:10.1007/s11537-012- 1140-8. (page 4)

    work page doi:10.1007/s11537-012- 2012

  22. [22]

    Remark on the Farey fraction spin chain

    Marc Technau. Remark on the Farey fraction spin chain. Proc. Am. Math. Soc. , 152(1):63–69, 2024. doi: 10.1090/proc/16520. (page 1)

    work page doi:10.1090/proc/16520 2024

  23. [23]

    A. V. Ustinov. Spin chains and Arnold’s problem on the Gauss-Kuz’min statistics for quadratic irrationals. Sb. Math., 204(5):762–779, 2013. URL: iopscience.iop.org/1064-5616/page/Abstracted, doi:10.1070/ SM2013v204n05ABEH004319. (page 1). (Jack Anderson) Department of Mathematics, University of Illinois at Urbana-Champaign, Ur- bana, IL 61801, USA. Email ...

    work page 2013