Zaremba Conjecture Overview

• Zaremba Conjecture is a statement in number theory and dynamics asserting every positive integer is the denominator of a continued fraction with all partial quotients bounded by an absolute constant, conjecturally 5.
• Recent advances using spectral theory, circle method, and combinatorial techniques have established density-one results and improved bounds for the set of eligible denominators.
• The conjecture has practical relevance in pseudorandom number generation and quasi-Monte Carlo methods, demonstrating its broad impact on discrete geometry and rational approximations.

The Zaremba Conjecture is a central open problem at the intersection of number theory, spectral theory, and the dynamics of thin orbits, with far-reaching connections to Diophantine approximation, pseudorandomness, and discrete geometry. It predicts a remarkable uniformity in the continued fraction expansions of rationals: every integer denominator should occur as the denominator of a finite continued fraction whose partial quotients are bounded by an absolute constant, conjecturally as small as $5$. Over the past decade, substantial advances have been made in both the proportion of integers for which the conjecture can be established and the methods by which such results are obtained.

1. Statement and Formulations

Zaremba's Conjecture, posed in 1971, asserts the existence of an absolute integer $A$ such that for every positive integer $q \geq 1$, there exists an integer $1 \leq p < q$ with $\gcd(p, q) = 1$ so that the regular continued fraction expansion

$\frac{p}{q} = [a_1, a_2, \ldots, a_k]$

satisfies $\max_j a_j \leq A$. The conjectural minimal value is $A=5$ (Bourgain et al., 2011, Dubno, 2023, Kontorovich, 2012).

The conjecture is typically formalized via the set

$$\mathcal{Q}_A = \Bigl\{ q \in \mathbb{N} \mid \exists\, p:\; (p,q)=1,\;\frac{p}{q} = [a_1,\ldots,a_k],\;\max_j a_j \leq A \Bigr\}$$

and predicts $\mathcal{Q}_A = \mathbb{N}$ for some $A$0.

A density-one version asks whether there exists $A$1 such that $A$2.

A related strong form asserts that for every $A$3, there is $A$4 with $A$5 and $A$6 (Dubno, 2023).

2. Quantitative and Density-One Progress

The full conjecture remains unresolved, but powerful quantitative methods have been brought to bear, resulting in the following principal breakthroughs:

• Density-one theorem: Bourgain and Kontorovich proved that for some large $A$7 (explicitly $A$8, improved to $A$9), the set $q \geq 1$0 has natural density one (Bourgain et al., 2011, Bourgain et al., 2011, Kontorovich, 2012). That is, almost all positive integers $q \geq 1$1 appear as denominators for bounded partial quotients:

$q \geq 1$2

• Improved numeric bounds: Through successive refinements exploiting spectral theory, expander graphs, and then elementary counting methods, the threshold $q \geq 1$3 for positive density was reduced to $q \geq 1$4 (Frolenkov et al., 2012), then to $q \geq 1$5 (Frolenkov et al., 2013), and most recently to $q \geq 1$6 (Kan, 2015, Kan, 2016). For these small values, one has a positive proportion:

$q \geq 1$7

• Explicit sequences: Via constructive algorithms leveraging the folding lemma (Mendes France, van der Poorten–Shallit), specific geometric families (such as $q \geq 1$8) were shown to satisfy Zaremba-type bounds with small $q \geq 1$9; recent developments extend these to more general geometric progressions (Dubno, 2023).
• General moduli with arithmetic conditions: For $1 \leq p < q$0 with radical $1 \leq p < q$1, Shulga established that $1 \leq p < q$2 admits an expansion with $1 \leq p < q$3 (exempting $1 \leq p < q$4 and $1 \leq p < q$5) (Shulga, 2023).

3. Theoretical and Methodological Foundations

Cantor-Like Structure and Hausdorff Dimension

For each $1 \leq p < q$6, define a Cantor-like set

$1 \leq p < q$7

whose Hausdorff dimension $1 \leq p < q$8 governs the richness of expansions. Hensley established: $1 \leq p < q$9 as $\gcd(p, q) = 1$0 (Bourgain et al., 2011, Pollicott et al., 2020).

Counting denominators up to $\gcd(p, q) = 1$1 among all convergents arising from $\gcd(p, q) = 1$2, one finds

$\gcd(p, q) = 1$3

but the leap to full density-one (i.e., $\gcd(p, q) = 1$4) is achieved only via advanced analytic tools.

Thin Orbits and Semigroup Dynamics

The set of denominators is realized as the $\gcd(p, q) = 1$5 coordinate in the orbit of the vector $\gcd(p, q) = 1$6 under the semigroup $\gcd(p, q) = 1$7 generated by $\gcd(p, q) = 1$8, $\gcd(p, q) = 1$9 (Bourgain et al., 2011, Bourgain et al., 2011, Kontorovich, 2012, Kontorovich, 2016). This "thin orbit" structure reflects deep arithmetic and geometric properties not present in full-group dynamics.

Circle Method and Exponential Sums

Key analytic input derives from the Hardy-Littlewood circle method, decomposing the unit circle into major arcs (well-approximated rationals of small denominator) and minor arcs. Exponential sums over the semigroup are analyzed: $\frac{p}{q} = [a_1, a_2, \ldots, a_k]$0 where $\frac{p}{q} = [a_1, a_2, \ldots, a_k]$1 is the denominator corresponding to $\frac{p}{q} = [a_1, a_2, \ldots, a_k]$2 (Bourgain et al., 2011). The density is expressed via: $\frac{p}{q} = [a_1, a_2, \ldots, a_k]$3 Minor-arc control uses multilinear Vinogradov-type bounds, and spectral gap techniques or thermodynamic formalism (transfer operator methods) supply the crucial major-arc asymptotics (Bourgain et al., 2011, Kontorovich, 2012, Kan, 2015, Frolenkov et al., 2012, Pollicott et al., 2020).

4. Extensions, Algorithms, and Explicit Constructions

Radical Bound and Arithmetic Refinements

For composite moduli, a radical bound is established: if $\frac{p}{q} = [a_1, a_2, \ldots, a_k]$4, then there exists $\frac{p}{q} = [a_1, a_2, \ldots, a_k]$5 coprime to $\frac{p}{q} = [a_1, a_2, \ldots, a_k]$6 such that in the continued fraction expansion of $\frac{p}{q} = [a_1, a_2, \ldots, a_k]$7, all partial quotients are at most $\frac{p}{q} = [a_1, a_2, \ldots, a_k]$8 (Shulga, 2023). This generalizes prior explicit results for $\frac{p}{q} = [a_1, a_2, \ldots, a_k]$9 (Niederreiter $\max_j a_j \leq A$0).

Combinatorial Folding Lemma and Algorithms

Constructive algorithms using the folding lemma can produce infinite families of powers of integers for which Zaremba's conjecture holds with $\max_j a_j \leq A$1 as small as $\max_j a_j \leq A$2. Dubno's refinement, for instance, gives that all $\max_j a_j \leq A$3 and $\max_j a_j \leq A$4 can be written as fractions with all partial quotients bounded by $\max_j a_j \leq A$5, with further generalization to composite bases (Dubno, 2023).

Uniform Bounds for Primes and Metric Results

Shkredov proved for all sufficiently large primes, there exists $\max_j a_j \leq A$6 coprime to $\max_j a_j \leq A$7 with maximal partial quotient $\max_j a_j \leq A$8, and a substantial fraction of numerators realize even sharper bounds (Shkredov, 14 Mar 2026). This enhances earlier results of Korobov ($\max_j a_j \leq A$9) and aligns with the metric theory of continued fractions, showing that for "most" numerators, partial quotients are typically small.

5. Connections, Applications, and Broader Impacts

Zaremba's conjecture has direct implications for pseudorandom number generation (e.g., low-discrepancy sequences for quasi-Monte Carlo methods) and the uniform distribution of linear congruential generators (Bourgain et al., 2011, Kontorovich, 2012, Kontorovich, 2016).

A significant connection exists to the periodic continued fraction problem in real quadratic fields (McMullen's conjecture). Mercat established that if Zaremba's conjecture holds for $A=5$0, then every real quadratic field contains infinitely many purely periodic continued fractions with partial quotients bounded by $A=5$1 (Mercat, 2016).

Spectral theory and the transfer operator framework, including quantitative Hausdorff dimension estimates, play a key role in current progress. The Pollicott–Vytnova approach provides sharp, fully rigorous bounds for dimension thresholds underpinning density results for small $A=5$2 (Pollicott et al., 2020).

6. Open Problems and Future Directions

• Proving the full Zaremba conjecture for all integers $A=5$3 and some absolute $A=5$4, especially $A=5$5, remains open. Lowering the required Hausdorff dimension threshold further, or finding fundamentally new techniques for the thin-orbit minor-arc problem, are pivotal outstanding challenges (Kontorovich, 2012, Pollicott et al., 2020).
• Explicit characterization of the exceptional set (if any) for small $A=5$6 and the role of local congruence obstructions is an active area.
• Automorphic and additive combinatorics approaches, enhanced spectral gap bounds, and potentially "local-global" methods for thin groups may push towards full resolution (Kontorovich, 2016, Kontorovich, 2012).
• The structural insights from Mahler functions and $A=5$7-regular sequences suggest further avenues for encoding and analyzing the set of bounded-quotient denominators, though no explicit new bounds for $A=5$8 arise from this perspective (Coons, 2017).

7. Summary Table of Density Results

Result	Value of $A=5$9	Density Statement	Methodology
Bourgain–Kontorovich (2011)	$$\mathcal{Q}_A = \Bigl\{ q \in \mathbb{N} \mid \exists\, p:\; (p,q)=1,\;\frac{p}{q} = [a_1,\ldots,a_k],\;\max_j a_j \leq A \Bigr\}$$0, $$\mathcal{Q}_A = \Bigl\{ q \in \mathbb{N} \mid \exists\, p:\; (p,q)=1,\;\frac{p}{q} = [a_1,\ldots,a_k],\;\max_j a_j \leq A \Bigr\}$$1	Full density: $$\mathcal{Q}_A = \Bigl\{ q \in \mathbb{N} \mid \exists\, p:\; (p,q)=1,\;\frac{p}{q} = [a_1,\ldots,a_k],\;\max_j a_j \leq A \Bigr\}$$2	Circle + spectral/transfer
Frolenkov–Kan, Kan–Frolenkov	$$\mathcal{Q}_A = \Bigl\{ q \in \mathbb{N} \mid \exists\, p:\; (p,q)=1,\;\frac{p}{q} = [a_1,\ldots,a_k],\;\max_j a_j \leq A \Bigr\}$$3	Positive density: $$\mathcal{Q}_A = \Bigl\{ q \in \mathbb{N} \mid \exists\, p:\; (p,q)=1,\;\frac{p}{q} = [a_1,\ldots,a_k],\;\max_j a_j \leq A \Bigr\}$$4	Elementary recurrences, combinatorics
Dubno algorithm	$$\mathcal{Q}_A = \Bigl\{ q \in \mathbb{N} \mid \exists\, p:\; (p,q)=1,\;\frac{p}{q} = [a_1,\ldots,a_k],\;\max_j a_j \leq A \Bigr\}$$5 (geometric)	All $$\mathcal{Q}_A = \Bigl\{ q \in \mathbb{N} \mid \exists\, p:\; (p,q)=1,\;\frac{p}{q} = [a_1,\ldots,a_k],\;\max_j a_j \leq A \Bigr\}$$6 for certain $$\mathcal{Q}_A = \Bigl\{ q \in \mathbb{N} \mid \exists\, p:\; (p,q)=1,\;\frac{p}{q} = [a_1,\ldots,a_k],\;\max_j a_j \leq A \Bigr\}$$7	Folding lemma
Shulga (radical bound)	$$\mathcal{Q}_A = \Bigl\{ q \in \mathbb{N} \mid \exists\, p:\; (p,q)=1,\;\frac{p}{q} = [a_1,\ldots,a_k],\;\max_j a_j \leq A \Bigr\}$$8	Explicit for arithmetic $$\mathcal{Q}_A = \Bigl\{ q \in \mathbb{N} \mid \exists\, p:\; (p,q)=1,\;\frac{p}{q} = [a_1,\ldots,a_k],\;\max_j a_j \leq A \Bigr\}$$9	Combinatorial, folding
Shkredov (primes)	$\mathcal{Q}_A = \mathbb{N}$0	All large primes	Cantor set, expansion

These results collectively advance towards the conjecture's full resolution and illuminate the intricate structure underlying rational approximation and discrete arithmetic dynamics.