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Spanning trees and continued fractions

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Pith's one-line read The set of possible numbers of spanning trees of connected simple planar graphs on n vertices grows exponentially with n, answering a 1969 question of Sedláček.

desk verdict Settles the 55-year-old exponential growth question with an elegant continued-fraction reduction; the stronger positive-proportion theorem rests on an uncertified numeric inequality that should be fixed. read the letter →

arxiv 2411.18782 v2 pith:6F3USD2K submitted 2024-11-27 math.CO math.NT

Swee Hong Chan , Alex Kontorovich , Igor Pak This is my paper · ORCID
classification math.COmath.NT MSC 05C3011A5511K5528A80
keywords spanningtreesplanargraphscontinuedfractionsZaremba'sconjecturethinorbitsHausdorffdimensiontransferoperatorexponentialgrowth
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that the set of possible numbers of spanning trees of connected simple planar graphs on $n$ vertices grows exponentially with $n$, settling a question raised by Sedláček in 1969. The proof connects graph theory to Diophantine approximation: certain continued fraction expansions are shown to produce graphs with prescribed spanning-tree counts, and a positive proportion of integers admit such expansions with bounded partial quotients. That Diophantine statement is proved using techniques developed for Zaremba's conjecture, and the decisive numerical ingredient is a rigorous computer-assisted estimate that a certain Cantor-like fractal has Hausdorff dimension above $0.775$. The argument reduces the former open problem to two explicit inequalities involving a single polynomial and a transfer operator, which the paper verifies directly.

What carries the argument

The machinery is a bridge from graphs to continued fractions and back. On the graph side, two operations on a marked planar graph, subdividing an edge ($\Phi_k$) and adding parallel edges ($\Psi_k$), multiply the spanning-tree vector $(\tau(G-e),\tau(G/e))$ by the shear matrices $\begin{pmatrix}1&k\\0&1\end{pmatrix}$ and $\begin{pmatrix}1&0\\k&1\end{pmatrix}$; composing these in alternating order produces exactly the matrix $\begin{pmatrix}*&*\\t&u\end{pmatrix}$ attached to the continued fraction $t/u=[b_1,1,b_2,1,\ldots,b_m,1]$. This yields the Main Graph Theorem: such a fraction gives a simple planar graph with $\tau(G)=t$ and $|V|=b_2+\cdots+b_m+2$. On the Diophantine side, the set $R_A$ of such fractions with $1\le b_i\le A$ is captured by the semigroup $\Gamma_A$ generated by the products $\begin{pmatrix}0&1\\1&1\end{pmatrix}\begin{pmatrix}0&1\\1&b\end{pmatrix}$, and its orbit under a fixed vector produces the numerators $t$. The orbital circle method, refined to threshold $\delta_0=0.775$, says that if the limit set of $\Gamma_A$ has Hausdorff dimension $\vartheta_A>\delta_0$, then a positive proportion of admissible integers occur as numerators. Finally, $\vartheta_A$ is controlled by the pressure zero of the transfer operator $L_s f(x)=\sum_{b=1}^A |T_b'(x)|^s f(T_b(x))$ with $T_b(x)=\frac{b+x}{1+b+x}$; a positive polynomial $f$ with $L_s f>f$ on $[0,1]$ proves $\vartheta_A>s$, and such an $f$ is exhibited for $s=0.775$ and $A=110$.

What would settle it

Run the Section 4.1 verification with rigorous interval arithmetic or a certified eigenvalue computation for the $5\times 5$ transfer matrix at $s=0.775$, $A=110$; if any $x\in[0,1]$ has $L_s f_s(x)-f_s(x)\le 0$, or if the computed top eigenvalue is at most $1$, then the dimension theorem is false.

Watch

Extended reading notes

Core claim

This paper claims that the number of distinct spanning-tree counts of connected simple planar graphs on $n$ vertices grows at least exponentially in $n$, and that a positive proportion of the integers up to that exponential scale are realized as such counts. The proof has four interlocking pieces. First, for a marked planar graph the spanning-tree vector $(\tau(G-e),\tau(G/e))$ is transformed by two operations, subdividing an edge and adding parallel edges, so that successive applications multiply the vector by shear matrices; reading the coordinates as numerator and denominator, the construction realizes exactly those rationals whose continued fraction expansion alternates $1$'s with positive digits, i.e. $t/u=[b_1,1,b_2,1,\ldots,b_m,1]$. Second, a Diophantine conjecture asserting that every integer $t$ admits such a fraction $t/u$ with digits bounded by $A$ is shown to hold for a positive proportion of $t$, conditional on a Hausdorff dimension threshold $\delta_0<1$. Third, that threshold is supplied by the orbital circle method refined to $\delta_0=0.775$. Fourth, the dimension condition is verified: for $A=110$, the fractal of these alternating continued fractions has Hausdorff dimension strictly above $0.775$, as witnessed by an explicit polynomial satisfying a transfer-operator inequality. Taken together, these steps yield the exponential growth theorems.

Load-bearing premise

The entire theorem rests on the claim, supported in Section 4.1 only by decimal values and a plot, that the polynomial $f_s(x)=0.0121844x^4-0.0513245x^3+0.116313x^2-0.225988x+0.526229$ satisfies $L_s f_s(x)-f_s(x)>7\times 10^{-5}$ on $[0,1]$ at $s=0.775$ and $A=110$; if that inequality fails, the dimension bound and the positive-proportion theorem collapse.

Editorial extensions

If this is right

  • The cardinality $|T(n)|$ of the set of spanning-tree counts of connected simple planar graphs on $n$ vertices is at least $c^n$ for some $c>1$ and all large $n$, settling Sedláček's 1969 lower-bound question.
  • A positive proportion of the integers $1,\ldots,c^n$ occur as spanning-tree counts of such graphs; the paper conjectures in Remark 1.18 that this can be improved from positive proportion to density one.
  • The dual function $\alpha(t)$, the minimum number of vertices needed to realize $t$ spanning trees, satisfies $\alpha(t)=O(\log t)$ for a positive proportion of $t$.
  • The same exponential lower bound holds for the larger family of all simple graphs, the first such bound for that family.
  • Under the paper's own method the base constant $c$ can be taken as $1.1103$, and no constant above the golden ratio $\varphi\approx 1.618$ can be obtained by this approach.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The transfer-operator certificate could be turned into a fully machine-checkable proof by running the same inequality test under interval arithmetic; the paper's margin of $7\times 10^{-5}$ is small but the method is systematic, so this is a computational rather than a conceptual gap.
  • If the density-one variant described in Remark 1.18 is carried out, the conclusion would strengthen to almost every integer up to $c^n$ being a spanning-tree count, implying a density-one version of the bound $\alpha(t)=O(\log t)$.
  • The graph–continued fraction correspondence hints that similar encodings could attack Sedláček-type problems for other graph families, such as regular graphs, wherever a Zaremba-type Diophantine statement can be proved for the relevant set of fractions.
  • The connection to spectra of Laplace operators mentioned in the final remarks suggests the exponential growth result may be interpretable as a statement about limit points of the spanning-tree spectral invariant $s(G)=\log \tau(G)/|G|$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies T(n), the set of spanning-tree counts of connected simple planar graphs on n vertices. It proves Theorem 1.1, that |T(n)| > c^n for some c > 1, answering Sedláček's 1969 question, and Theorem 1.3, that a positive proportion of {1, ..., c^n} is realized as spanning-tree counts. The proof combines four ingredients: a Main Graph Theorem (Theorem 1.9) constructing simple planar graphs from continued fractions, a dualization to the function α(t), a use of Bourgain–Kontorovich-type thin-orbit technology to prove a positive-proportion form of Zaremba's conjecture (Theorems 1.12, 1.16), and a numerical Hausdorff-dimension lower bound (Theorem 1.19). The paper also gives an independent route to Theorem 1.1 in §3.2 that does not need the full positive-proportion result.

Significance. If the numerical step is made rigorous, this is a substantial advance: it settles a problem open since 1969 and proves a quantitatively stronger positive-proportion theorem. The graph-theoretic core (Theorem 1.9 and Lemmas 2.1–2.3) is elementary, explicit, and convincing, and the paper is transparent about its dependence on the deep external results of Bourgain–Kontorovich and Kan. The paper also includes concrete constants, a follow-up comparison in [ABG25], and useful discussion of related conjectures. The main obstacle is the uncertified numerical inequality in §4.1. This is a load-bearing step for Theorems 1.12, 1.7, and 1.3, and currently the proof is not complete as written. The issue is local and appears fixable with a rigorous interval-arithmetic certificate or exact rational enclosure.

major comments (3)
  1. [Section 4.1, inequalities following Eq. (4.4)] The proof of Theorem 1.19 is incomplete as written. The paper asserts that for s = 0.775, A = 110, and the polynomial f_s in (4.4), one has f_s(x) > 0.3 and (L_s f_s − f_s)(x) > 7×10^{-5} for every x in [0,1], citing Figure 4.1. Since L_s is a sum of 110 terms involving the exponent s = 31/40, the displayed 7-decimal coefficients and a plot do not constitute a proof of a strict pointwise inequality on a continuum. The margin 7×10^{-5} is small relative to what is needed to rule out rounding error without an enclosure. Because Theorem 1.17 requires ϑ_A > 0.775, Theorems 1.12, 1.7, and 1.3 all depend on this uncertified check. The later citation in §5.9 of Pollicott's 20-digit computation does not repair the gap, since that computation is also not certified in this paper. Please provide a rigorous interval-arithmetic or exact rational certificate, with code or data sufficient to verify (4.3).
  2. [Section 3.2, Eq. (3.4)] The passage from the counting estimate |B_N| = N^{2ϑ_A+o(1)} to the claimed lower bound |N_A ∩ [1,N]| ≫ N^{ϑ_A−o(1)} is not justified in the text. The argument that R_A also contains (t+u)/(t+2u) shows that each fraction t/u produces a new numerator t+u, but the sentence 't+u and t together determine the pair (t,u)' is a statement about ordered pairs and does not bound the number of old fractions that can have the same new numerator. As written, the multiplicity of a fixed n = t+u could in principle be large. Please either supply the multiplicity bound or cite the standard dimension/projection result behind this step. The same comment applies to the claim in Remark 4.2 that ϑ_A > 1/2 already for A = 4; if Theorem 1.1 is to be independent of the A = 110 certificate, a rigorous lower bound for ϑ_4 must be provided.
  3. [Section 3.3, Theorem 3.2 and Lemma 3.3] The adaptation of the Bourgain–Kontorovich theorem to the semigroup Γ_A needs to be made explicit. The paper quotes Theorem 3.2 as a general black box but does not state its full hypotheses, and Lemma 3.3 verifies only the congruence property Γ_A mod q = SL(2,Z/qZ). To apply the theorem to Γ_A, the authors should identify precisely which theorem from [BK14] is being used, confirm that Γ_A satisfies its hypotheses, and explain why the Hausdorff dimension of the semigroup's limit set is the quantity ϑ_A defined through C_A. I do not doubt that these checks can be done, but as written the route from [BK14, Theorem 1.8] and [Kan21] to Theorem 1.16 is a sketch rather than a verification.
minor comments (4)
  1. [Statement of Theorem 1.14] The name 'Zarembra' is a typo for 'Zaremba'.
  2. [References] The reference entry for [ABG25] contains a stray duplicated line 'Random Structures in Algorithms, 1 (1990), 175–181.' from the entry for [Alo90].
  3. [Section 3.4] The phrase 'Cauchy–Schwartz' should be 'Cauchy–Schwarz'.
  4. [Figures 4.1 and 4.2] The captions should state how the plotted values were computed and should not be used as evidence for the pointwise inequalities; the relevant inequalities need the rigorous certificate requested in the first major comment.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; the main derivation is self-contained apart from independently published inputs, and the numerical dimension check is a rigor caveat rather than a circular step.

full rationale

The claimed derivation does not reduce to its inputs. Theorem 1.1 is obtained in Section 3.2 directly from the explicit graph construction (Theorem 1.9, proved in Section 2 from deletion-contraction and the matrix-continued-fraction correspondence) plus a counting bound that needs only the fact that the Hausdorff dimension is positive, which the paper notes is elementary for A=2. The stronger Theorem 1.3 rests on the positive-proportion Diophantine theorem imported as Theorem 1.14 and Theorem 1.16 from the published independent works of Bourgain-Kontorovich and Kan; the threshold delta_0 = 0.775 is an external result of Kan, not derived in this paper. The only place where numerical computation is load-bearing is Theorem 1.19 and Section 4.1, where the paper produces a specific polynomial f_s and asserts the pointwise inequality L_s f_s - f_s > 7e-5 on [0,1]. This is an explicit witness check, not a fitted parameter renamed as a prediction; if the inequality holds, Lemma 4.1 gives the conclusion by a standard Ruelle-Perron-Frobenius argument. The self-citations [BK14] and [CP24c] do not create circularity because [BK14] is an independent prior theorem and Theorem 1.9 is proved in the text. The genuine weakness is rigor and reproducibility: the Section 4.1 verification is asserted from a plot with decimal coefficients and no interval-arithmetic certificate, so the completeness of Theorem 1.19 depends on an uncertified numerical claim; this is a correctness concern, not circularity.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new particles or entities. Its free parameters are computational choices in the witness-based dimension proof, not fitted physical or empirical constants. The main external assumptions are the Bourgain-Kontorovich thin orbit theorem and Kan's refined threshold, both published results that the paper uses as black boxes, together with standard transfer operator facts.

free parameters (2)
  • A=110 = 110
    Chosen as the bound on continued fraction partial quotients for the dimension check. The paper states A=100 fails, so this value is selected by hand to make the numerical inequality work.
  • N=5 = 5
    Order of Chebyshev-Lagrange interpolation used to construct the test polynomial for the transfer operator; chosen because it succeeds in producing a positive test function.
assumptions (6)
  • standard math Matrix-tree theorem: the number of spanning trees of a graph equals a determinant of a Laplacian cofactor matrix.
    Invoked for the determinant interpretation and for bounds on spanning tree counts; treated as background.
  • standard math Spanning tree deletion-contraction identity: tau(G)=tau(G-e)+tau(G/e).
    Used in Lemma 2.1 and throughout the graph construction.
  • standard math Ruelle-Perron-Frobenius theory: the transfer operator L_s has a unique maximal eigenvalue exp(P(s)), and the zero of P(s) gives the Hausdorff dimension of the continued fraction Cantor set.
    Invoked in Section 4.1 to connect pressure functions and transfer operator inequalities to dimension estimates; relies on [Rue82] and [PV22].
  • standard math Pollicott-Vytnova criterion: if a positive function f satisfies L_s f > f then the Hausdorff dimension exceeds s.
    Used in Lemma 4.1 as the rigorous route from the explicit polynomial to the dimension lower bound.
  • domain assumption Bourgain-Kontorovich local-global principle for thin orbits: if the Hausdorff dimension of the limit set exceeds a threshold, the orbit contains a positive proportion of admissible integers.
    Theorem 3.2 is a deep external result from [BK14] that carries the main Diophantine weight; it is not proved in this paper.
  • domain assumption Kan's improvement of the minor arcs analysis gives the threshold delta_0=0.775.
    Theorem 1.17 is quoted from [Kan21]; the paper does not reproduce its proof but bases the whole numerical threshold on it.

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Pith. "Pith review of Spanning trees and continued fractions." pith.science (2026). https://pith.science/paper/6F3USD2K

@misc{pith2026241118782,
  author       = {Pith},
  title        = {Pith review of: Spanning trees and continued fractions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6F3USD2K}},
  note         = {Machine review of arXiv:2411.18782}
}
abstract

We prove the exponential growth of the cardinality of the set of numbers of spanning trees in simple (and planar) graphs on $n$ vertices, answering a question of Sedl\'a\v{c}ek from 1969. The proof uses a connection with continued fractions, ``thin orbits,'' and Zaremba's conjecture.

Figures

Figures reproduced from arXiv: 2411.18782 by the authors.

Figure 1.1
Figure 1.1. A typical graph constructed in the Main Graph Theorem 1.9 Conjecture 1.10 (Diophantine Conjecture). There is a universal constant A > 0 so that, for every integer t ≥ 3, there is a coprime integer u > t, such that the quotient t/u has continued fraction expansion (1.7) with b1, . . . , bm ≤ A. We discuss in §5.7 why this conjecture may be plausible. Regardless, together with the Main Graph Theorem 1.9, it would sett… view at source ↗
Figure 1.2
Figure 1.2. The fractal C¯ defined in (1.9). Diameters of circles along the unit interval correspond to ranges being removed at each stage. A key role is played here by the Cantor-like sets comprising the limit points of these Diophantine fractions. In the setting of Zaremba’s conjecture, the limit set, for a given A > 0, is the following: FA :=  [a1, a2, . . .] : 1 ≤ ai ≤ A, ∀ i [PITH_FULL_IMAGE:figures/full_fig_p005_1_2.png] view at source ↗
Figure 2.1
Figure 2.1. Marked graphs (G, e), (G′ , e′ ) = Φ2 (G, e), (G′′, e′′) = Ψ3 (G, e) and (H, f) = Υ2 (G, e) [PITH_FULL_IMAGE:figures/full_fig_p007_2_1.png] view at source ↗
Figures from the paper (4 more)
Figure 2.2
Figure 2.2. Figure 2.2: Marked graph (G, e) = Υ3 Υ 1 Υ 2 Υ 4 (P1, u). From above, G = (V, E) is a connected simple planar graph. By induction, we have: |V | = b1 + . . . + bm + 2. Similarly, equation (2.3) gives by induction: (2.4) v(G, e) = (0, 1) ·  0 1 1 1   0 1 1 bm  · · ·  0 1 1 1…
Figure 2.3
Figure 2.3. Figure 2.3: A typical graph G − e from Lemma 2.3 This graph leaves “tails” of paths, which can be trimmed without changing the number of spanning trees, resulting in the graph shown in [PITH_FULL_IMAGE:figures/full_fig_p009_2_3.png]
Figure 4.1
Figure 4.1. Figure 4.1: (a) The test function fs in (4.4), and (b) difference Lsfs − fs on [0, 1]. For x ∈ [0, 1], it can be verified that fs(x) > 0.3, see [PITH_FULL_IMAGE:figures/full_fig_p015_4_1.png]
Figure 4.2
Figure 4.2. Figure 4.2: (a) The test function fs in (4.6), and (b) the difference Lsfs − fs. (4.6) fs(x) = 0.0123381x 4 − 0.0517567x 3 + 0.116202x 2 − 0.221186x + 0.524143. This function exceeds 0.3 on [0, 1], and Lsfs − fs is less than −0.0002 on [0, 1], see [PITH_FULL_IMAGE:figures/full_…

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On some results of Korobov and Larcher and Zaremba's conjecture

    math.NT 2026-03 accept novelty 7.0 of 10

    Zaremba's conjecture holds for all large primes: an absolute M exists so every large prime p admits a coprime a whose continued-fraction partial quotients are all ≤ M, with quantitative lower bounds on the number of such a.

  2. Effective resistance in planar graphs and continued fractions

    math.CO 2025-05 accept novelty 7.0 of 10

    For every rational resistance c/t, a simple planar graph with O(max(t/c, t/(t-c), log t)) vertices exists, and no graph can do better up to a constant.

Reference graph

Works this paper leans on

63 extracted references · 53 canonical work pages · cited by 2 Pith papers

  1. [1]

    Noga Alon, Spanning trees in regular graphs, Random Structures in Algorithms, 1 (1990), 175--181

    work page 1990

  2. [2]

    Random Structures in Algorithms, 1 (1990), 175--181

    Noga Alon, Matija Buci\'c and Lior Gishboliner, The spanning tree spectrum: improved bounds and simple proofs, preprint (2025), 8 pp.; arXiv:2503.23648 . Random Structures in Algorithms, 1 (1990), 175--181

    arXiv 2025

  3. [3]

    Jernej Azarija, Counting graphs with different numbers of spanning trees through the counting of prime partitions, Czechoslovak Math. J. 64 (2014), 31--35

    work page 2014

  4. [4]

    Jernej Azarija and Riste S krekovski, Euler's idoneal numbers and an inequality concerning minimal graphs with a prescribed number of spanning trees, Math. Bohem. 138 (2013), 121--131

    work page 2013

  5. [5]

    15 (2010), 81--86

    Thomas Bier, Formulas for the number of spanning trees in a chain of cycles, SQU Jour.\ Sci. 15 (2010), 81--86

    work page 2010

  6. [6]

    22 (2006), 185--202

    Nicolas Bonichon, Cyril Gavoille, Nicolas Hanusse, Dominique Poulalhon and Gilles Schaeffer, Planar graphs, via well-orderly maps and trees, Graphs Combin. 22 (2006), 185--202

    work page 2006

  7. [7]

    Jonathan Borwein, Alf van der Poorten, Jeffrey Shallit and Wadim Zudilin, Neverending fractions, Cambridge Univ.\ Press, Cambridge, UK, 2014, 212 pp

    work page 2014

  8. [8]

    24 (2018), 2831--2839

    Jean Bourgain, On quadratic irrationals with bounded partial quotients, Selecta Math. 24 (2018), 2831--2839

    work page 2018

Show all 63 references
  1. [9]

    Rufus Bowen, Hausdorff dimension of quasicircles, Publ.\ IHES 50 (1979), 11--25

    1979

  2. [10]

    180 (2014), 137--196

    Jean Bourgain and Alex Kontorovich, On Zaremba's conjecture, Ann.\ of Math. 180 (2014), 137--196

    2014

  3. [11]

    2018 (2018), Paper No

    Jean Bourgain and Alex Kontorovich, Beyond expansion IV: Traces of thin semigroups, Discrete Anal. 2018 (2018), Paper No. 6, 27 pp

    2018

  4. [12]

    Kevin Buchin and Andr\'e Schulz, On the number of spanning trees a planar graph can have, in Proc.\ 18th ESA, Springer, Berlin, 2010, 110--121

    2010

  5. [13]

    Bumby, Hausdorff dimension of sets arising in number theory, in Lecture Notes in Math

    Richard T. Bumby, Hausdorff dimension of sets arising in number theory, in Lecture Notes in Math. 1135, Springer, Berlin, 1985, 1--8

    1985

  6. [14]

    122 (2024), 104018, 14 pp

    Swee Hong Chan and Igor Pak, Linear extensions and continued fractions, European J.\ Combin. 122 (2024), 104018, 14 pp

    2024

  7. [15]

    1015 (2024), Paper No

    Swee Hong Chan and Igor Pak, Computational complexity of counting coincidences, Theor.\ Comp.\ Sci. 1015 (2024), Paper No. 114776

    2024

  8. [16]

    Swee Hong Chan and Igor Pak, Equality cases of the Stanley–Yan log-concave matroid inequality, preprint (2024), 35 pp.; arXiv:2407.19608

    2024 arXiv

  9. [17]

    Frolenkov and Igor D

    Dmitrii A. Frolenkov and Igor D. Kan, A strengthening of a theorem of Bourgain--Kontorovich II, Mosc.\ J.\ Comb.\ Number Theory 4 (2014), 78--117

    2014

  10. [18]

    Good, The fractional dimensional theory of continued fractions, Proc.\ Cambridge Philos.\ Soc

    Irving J. Good, The fractional dimensional theory of continued fractions, Proc.\ Cambridge Philos.\ Soc. 37 (1941), 199--228; Corrigenda: 105 (1989), 607

    1941

  11. [19]

    Grimmett, An upper bound for the number of spanning trees of a graph, Discrete Math

    Geoffrey R. Grimmett, An upper bound for the number of spanning trees of a graph, Discrete Math. 16 (1976), 323--324

    1976

  12. [20]

    48 (1947), 966--993

    Marshall Hall, On the sum and product of continued fractions, Annals Math. 48 (1947), 966--993

    1947

  13. [21]

    62 (2023), 69--110

    Jaroslav Han c l and Ond r ej Turek, Continued fractions with bounded even-order partial quotients, Ramanujan J. 62 (2023), 69--110

    2023

  14. [22]

    Hardy and Edward M

    Godfrey H. Hardy and Edward M. Wright, An introduction to the theory of numbers (sixth ed., revised), Oxford Univ.\ Press, Oxford, 2008, 621 pp

    2008

  15. [23]

    Doug Hensley, The distribution of badly approximable numbers and continuants with bounded digits, in Th\'eorie des nombres, de Gruyter, Berlin, 1989, 371--385

    1989

  16. [24]

    Number Theory 40 (1992), 336--358

    Doug Hensley, Continued fraction Cantor sets, Hausdorff dimension, and functional analysis, J. Number Theory 40 (1992), 336--358

    1992

  17. [25]

    Number Theory 58 (1996), 9--45

    Doug Hensley, A polynomial time algorithm for the Hausdorff dimension of continued fraction Cantor sets, J. Number Theory 58 (1996), 9--45

    1996

  18. [26]

    Doug Hensley, Continued fractions, World Sci., Hackensack, NJ, 2006, 245 pp

    2006

  19. [27]

    25 (2015), 860--914

    ShinnYih Huang, An improvement to Zaremba's conjecture, Geom.\ Funct.\ Anal. 25 (2015), 860--914

    2015

  20. [28]

    4 (2004), 63--76

    Oliver Jenkinson, On the density of Hausdorff dimensions of bounded type continued fraction sets: the Texan conjecture, Stoch.\ Dyn. 4 (2004), 63--76

    2004

  21. [29]

    Oliver Jenkinson and Mark Pollicott, Computing the dimension of dynamically defined sets: E_2 and bounded continued fractions, Ergodic Theory Dynam.\ Systems 21 (2001), 1429--1445

    2001

  22. [30]

    325 (2018), 87--115

    Oliver Jenkinson and Mark Pollicott, Rigorous effective bounds on the Hausdorff dimension of continued fraction Cantor sets: a hundred decimal digits for the dimension of E_2 , Adv.\ Math. 325 (2018), 87--115

    2018

  23. [31]

    Oliver Jenkinson and Mark Pollicott, Rigorous dimension estimates for C antor sets arising in Z aremba theory, In Dynamics: topology and numbers. Contemp. Math. 744 (2020), 83--107

    2020

  24. [32]

    Kan, A strengthening of a theorem of Bourgain and Kontorovich III, Izv.\ Math

    Igor D. Kan, A strengthening of a theorem of Bourgain and Kontorovich III, Izv.\ Math. 79 (2015), 288--310

    2015

  25. [33]

    Kan, A strengthening of a theorem of Bourgain and Kontorovich IV, Izv.\ Math

    Igor D. Kan, A strengthening of a theorem of Bourgain and Kontorovich IV, Izv.\ Math. 80 (2016), 1094--1117

    2016

  26. [34]

    Kan, A strengthening of a theorem of Bourgain and Kontorovich V, Proc.\ Steklov Inst.\ Math

    Igor D. Kan, A strengthening of a theorem of Bourgain and Kontorovich V, Proc.\ Steklov Inst.\ Math. 296 (2017), 125--131

    2017

  27. [35]

    Kan, A strengthening of the Bourgain--Kontorovich method: three new theorems, Sb.\ Math

    Igor D. Kan, A strengthening of the Bourgain--Kontorovich method: three new theorems, Sb.\ Math. 212 (2021), 921--964

    2021

  28. [36]

    Knuth, The art of computer programming.\ Vol

    Donald E. Knuth, The art of computer programming.\ Vol. 2.\ Seminumerical algorithms (third ed.), Addison-Wesley, Reading, MA, 1998, 762 pp

    1998

  29. [37]

    Alex Kontorovich, From Apollonius to Zaremba: local-global phenomena in thin orbits, Bull.\ AMS 50 (2013), 187--228

    2013

  30. [38]

    AMS 30 (2017), 1023--1046

    Alex Kontorovich, Peter McNamara and Geordie Williamson, Appendix to: S chubert calculus and torsion explosion (by G eordie W illiamson), Jour. AMS 30 (2017), 1023--1046

    2017

  31. [39]

    163 (1989), 1--55

    Steven Lalley, Renewal theorems in symbolic dynamics, with applications to geodesic flows, non-Euclidean tessellations and their fractal limits, Acta Math. 163 (1989), 1--55

    1989

  32. [40]

    101 (1986), 135--150

    Gerhard Larcher, On the distribution of sequences connected with good lattice points, Monatsh.\ Math. 101 (1986), 135--150

    1986

  33. [41]

    McKay, Spanning trees in regular graphs, European J.\ Combin

    Brendan D. McKay, Spanning trees in regular graphs, European J.\ Combin. 4, (1983), 149--160

    1983

  34. [42]

    With an appendix by Jean Bourgain, Alex Kontorovich and Michael Magee, J

    Michael Magee, Hee Oh and Dale Winter, Uniform congruence counting for Schottky semigroups in _2( ) . With an appendix by Jean Bourgain, Alex Kontorovich and Michael Magee, J. Reine Angew.\ Math. 753 (2019), 89--135

    2019

  35. [43]

    98 (1973), 95--97

    Ladislav Nebesk\'y, On the minimum number of vertices and edges in a graph with a given number of spanning trees, C asopis P e st.\ Mat. 98 (1973), 95--97

    1973

  36. [44]

    101 (1986), 309--315

    Harald Niederreiter, Dyadic fractions with small partial quotients, Monatsh.\ Math. 101 (1986), 309--315

    1986

  37. [45]

    Marc Noy, Graphs, in Handbook of enumerative combinatorics, CRC Press, Boca Raton, FL, 2015, 397--436

    2015

  38. [46]

    Mark Pollicott, Dimension of points with restricted even order partial quotients, preprint (2025), 10 pp; available at tinyurl.com/y3x8nbhe https://warwick.ac.uk/fac/sci/maths/people/staff/mark_pollicott/p3/ckp.pdf

    2025

  39. [47]

    B 9 (2022), 1102--1159

    Mark Pollicott and Polina Vytnova, Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups, Trans.\ AMS, Ser. B 9 (2022), 1102--1159

    2022

  40. [48]

    Ares Rib\'o Mor, Realization and counting problems for planar structures, Dissertation, Freie Universit\"at Berlin, 2006, 192 pp

    2006

  41. [49]

    James Rickards and Katherine E. Stange, Reciprocity obstructions in semigroup orbits in (2, ) , preprint (2024), 28 pp.; arXiv:2401.01860 ; supplement available on GitHub at tinyurl.com/ys8h9p3m https://github.com/JamesRickards-Canada/Semigroup-Reciprocity

    2024

  42. [50]

    David Ruelle, Repellers for real analytic maps, Ergodic Theory Dynam.\ Systems 2 (1982), 99--107

    1982

  43. [51]

    Rukavishnikova, The law of large numbers for the sum of partial quotients of a rational number with a fixed denominator, Math.\ Notes 90 (2011), 418--430

    Maria G. Rukavishnikova, The law of large numbers for the sum of partial quotients of a rational number with a fixed denominator, Math.\ Notes 90 (2011), 418--430

    2011

  44. [52]

    Peter Sarnak, Prescribing the spectra of locally uniform geometries, Chern Lectures, Berkeley, CA, 2023; available at publications.ias.edu/node/2728 https://publications.ias.edu/sarnak/paper/2728

    2023

  45. [53]

    Ji r \'i Sedl\'a c ek, On the spanning trees of finite graphs (in Czech), C asopis P e st. Mat. 91 (1966), 221--227

    1966

  46. [54]

    Ji r \'i Sedl\'a c ek, On the number of spanning trees of finite graphs, C asopis P e st. Mat. 94 (1969), 217--222

    1969

  47. [55]

    13 (1970), 515--517

    Ji r \'i Sedl\'a c ek, On the minimal graph with a given number of spanning trees, Canad.\ Math.\ Bull. 13 (1970), 515--517

    1970

  48. [56]

    645 (2022), 229--236

    Rikhav Shah, Determinants of binary matrices achieve every integral value up to (2^n / n) , Linear Algebra Appl. 645 (2022), 229--236

    2022

  49. [57]

    Nikita Shulga, Radical bound for Zaremba's conjecture, Bull.\ LMS 56 (2024), 2615--2624

    2024

  50. [58]

    Neil J. A. Sloane, The O nline E ncyclopedia of I nteger S equences, oeis.org http://oeis.org

  51. [59]

    82 (2022), 182--196

    Richard Stong, Minimal graphs with a prescribed number of spanning trees, Australas.\ J.\ Combin. 82 (2022), 182--196

    2022

  52. [60]

    Tutte, Graph theory, Addison-Wesley, Reading, MA, 1984, 333 pp

    William T. Tutte, Graph theory, Addison-Wesley, Reading, MA, 1984, 333 pp

    1984

  53. [61]

    Vu, Recent progress in combinatorial random matrix theory, Probab.\ Surv

    Van H. Vu, Recent progress in combinatorial random matrix theory, Probab.\ Surv. 18 (2021), 179--200

    2021

  54. [62]

    Yao and Donald E

    Andrew C. Yao and Donald E. Knuth, Analysis of the subtractive algorithm for greatest common divisors, Proc.\ Nat.\ Acad.\ Sci.\ USA 72 (1975), 4720--4722

    1975

  55. [63]

    Stanis aw K. Zaremba, La m\'ethode des ``bons treillis'' pour le calcul des int\'egrales multiples (in French), in Applications of number theory to numerical analysis, Academic Press, New York, 1972, 39--119

    1972

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