Zaremba's Conjecture in Diophantine Approximation

• Zaremba’s Conjecture is an open problem in Diophantine approximation that posits the existence of a uniform bound for the partial quotients in the continued fraction expansion of any rational number.
• Breakthroughs combining analytic, combinatorial, and dynamical methods have led to density-one and positive proportion results with bounded alphabets, reducing the threshold to values as low as 4 or 5.
• Recent extensions explore non-classical settings such as p-adic and complex continued fractions, revealing the conjecture’s deep interdisciplinary impact and stimulating further algorithmic and structural research.

Zaremba's Conjecture is a central open problem in the theory of Diophantine approximation and continued fractions, addressing the uniform boundedness of partial quotients in the continued fraction expansion of rationals with arbitrary denominators. It has motivated deep interactions between analytic number theory, dynamics, thin groups, combinatorics, and more recently, connections to higher-dimensional and non-classical settings.

1. Statement and Formulation of Zaremba's Conjecture

Let $q\ge 2$ be an integer. For $1\leq a<q$ with $\gcd(a,q)=1$, the simple continued fraction expansion of $a/q$ is

$$\frac{a}{q} = [a_1,\ldots,a_n],\quad a_j\in \mathbb{N}$$

meaning

$$\frac{a}{q} = \cfrac{1}{a_1+\cfrac{1}{a_2+\cdots+\cfrac{1}{a_n}}}.$$

Set $K(a/q) := \max\{a_1,a_2,\ldots,a_n\}$.

Zaremba's Conjecture (1971): There exists an absolute constant $t\ge 1$ such that for every integer $q\ge 2$, there is $1\le a<q$, $\gcd(a,q)=1$, with $K(a/q)\le t$. Zaremba conjectured $t=5$.

Equivalent "alphabet" formulation: For some $A$, every $q$ is the denominator of a rational number $a/q$ whose continued fraction expansion has all partial quotients in $\{1,2,\ldots,A\}$.

No smaller value than $t=2$ is possible for all $q$; for instance, $1/q=[1,1,\ldots,1]$ corresponds to $q$ a Fibonacci number only. For primes $q$, even tighter bounds are conjectured for “almost all” $q$ (Shulga, 2023).

2. Historical Development and Key Partial Results

Zaremba's problem has a rich history, with milestones and improvements along several lines:

• Korobov (1963): For all $q$, some $a$ exists with $K(a/q)\ll \log q$.
• Niederreiter (1986): For $q=2^n$ or $3^n$, $K(a/q)\le 3$, and for $q=5^n$, $K(a/q)\le 4$.
• Yodphotong–Laohakosol (2002): For $q=6^n$, $K(a/q)\le 5$.
• Bourgain–Kontorovich (2011): For $A=50$, almost all $q$ (density one subset) are attained as denominators with $K(a/q)\le A$ (Bourgain et al., 2011).
• Frolenkov–Kan, Huang, Kan: Reduced thresholds for "positive density" or "density-one" results to $A=5$ and ultimately $A=4$ (Kan, 2014, Kan, 2015, Kan, 2016, Huang, 2013).
• Moshchevitin–Murphy–Shkredov (2022): For large $q$ with large minimal prime factors, $K(a/q)\ll \log q/\log\log q$; for $q=p^n$, $p$ fixed, $n\gg 1$, same bound holds (Moshchevitin et al., 2022).

A table of key results for reference:

Authors	$q$ type	Bound on $K(a/q)$
Korobov	all $q$	$O(\log q)$
Niederreiter	$2^n$, $3^n$	$3$
Niederreiter	$5^n$	$4$
Yodphotong–Laohakosol	$6^n$	$5$
Frolenkov–Kan (elementary methods)	positive density; $q$ arbitrary	$5$
Bourgain–Kontorovich, Huang, Kan, etc.	density one; $q$ arbitrary	$50 \downarrow 5$
Kan	positive density; $A=4$	$4$
Moshchevitin, Murphy, Shkredov	$q$ with large prime factors	$O(\log q/\log\log q)$
Shulga	$q\neq 2^n,3^n$	$\mathrm{rad}(q)-1$

3. The Radical Bound and Constructive Combinatorics

The "radical bound" is a constructive, combinatorial improvement due to Shulga (Shulga, 2023). For any $q\ge 2$, not a power of $2$ or $3$,

$K(a/q) \le \mathrm{rad}(q)-1$

with $\mathrm{rad}(q)=\prod_{p|q} p$ (the radical of $q$).

Notable consequences:

• For $q=2^n3^m$, $\mathrm{rad}(q)=6$, so $K(a/q)\le 5$, covering all such mixed powers and generalizing previous results for $q=2^n$ and $q=6^n$.
• For $q=p^n$, $K(a/q)\le p-1$, which sharpens the analytic $O(\log q/\log\log q)$ bound when $n\gg p^2$.

The construction uses the classical "Folding Lemma" for continued fractions. The proof iteratively peels off prime factors of $q$, maintaining boundedness of partial quotients by "folding" at each stage, starting from a squarefree base case and reconstructing $a/q$ via a sequence of controlled combinatorial operations. No deep analytic machinery is invoked; it is entirely explicit.

4. Density-One and Positive-Proportion Results

The analytic–combinatorial breakthrough was achieved by Bourgain–Kontorovich, who proved for some large $A$ (originally $A=2189$, then $A=50$), all but a zero density subset of positive integers $q$ are attainable as denominators with $K(a/q)\le A$ (Bourgain et al., 2011). Further advances lowered $A$ for positive-proportion and density-one results:

• Frolenkov–Kan, Huang, Kan (2012–2015): By refined exponential sum bounds and the construction of quasi-independent matrix "ensembles" in $\mathrm{SL}_2(\mathbb{Z})$, the threshold for density one was pushed down to $A=5$ (numerically, the minimal $A$ so that the Hausdorff dimension of $[0; a_1,a_2,\ldots]$ with $a_j\leq A$ exceeds $5/6$) (Huang, 2013, Kan, 2014, Kan, 2015, Kan, 2016). Positive-proportion was established for $A=4$.
• Partial Proportion for Sparse Alphabets: Non-consecutive small alphabets (e.g., $\{1,2,3,4,10\}$) yield positive proportion with slightly weaker dimension thresholds (Kan, 2014).
• Hausdorff Dimension Paradigm: The central metric is $\delta_A = \dim_H(C_A)$, the Hausdorff dimension of the bounded-digit Cantor set. For positive density, $\delta_A>0.7807\dots$ suffices; for density one, $\delta_A>5/6$ (Kan, 2016, Huang, 2013).

5. Algorithmic, Structural, and Non-Archimedean Extensions

Explicit algorithms, particularly those using the folding lemma in a structured way, produce infinite families of "explicit Zaremba sequences" (i.e., geometric progressions of $q$ with $K(a/q)\le A$). For instance, the refined folding algorithm in (Dubno, 2023) produces families of the form $12^k$ or $18^k$ with $A=5$, improving classical approaches by better handling of non-squarefree bases.

Further, matrix semigroup and dynamical perspectives, especially via affine sieve techniques and thermodynamic formalism, have been crucial for understanding the "randomness" and multiplicity of denominators with bounded partial quotients (Cohen, 2016, Coons, 2017). The semigroup $\Gamma_A$ generated by

$$M_a = \begin{pmatrix} 0 & 1 \ 1 & a \end{pmatrix},\quad a\in\{1,2,\ldots,A\}$$

models the combinatorics of continued fraction constructions, and its orbits encode the denominators' structure.

6. Generalizations and Analogues in Other Contexts

Extensions to complex and $p$-adic continued fractions are under active paper. The Hurwitz–Zaremba analogue for Gaussian integers considers whether, for all $q\in\mathbb{Z}[i]$, there exists $r\in\mathbb{Z}[i]$, $\gcd(r,q)=1$ so that $r/q$ admits a Hurwitz continued fraction expansion with bounded Gaussian-integer partial quotients (Robert et al., 2023). Specialized sequences over $\mathbb{Z}[i]$ (e.g., $q=(-3\pm i)^k$) have been established with explicit bounds, but the general case remains open.

Recent work has shown asymptotic “on average” support for Zaremba’s conjecture in complex quadratic fields, applying transfer operator methods for fractal sets of bounded type (Lee, 12 Dec 2025).

7. Open Problems and Current Directions

• Uniform Constant for All $q$: Despite major advances, the original conjecture—i.e., $K(a/q)\le 5$ for all $q$—remains open. The "radical bound" (Shulga, 2023) and density-one results hold in various broad classes, but the exceptional set is not fully eliminated.
• Further Lowering of $A$: Whether the positive density and density-one results can be achieved for $A=3$ or $A=2$ is a major open question.
• Distribution and Multiplicity: Precise asymptotics for the number of representations ("multiplicity") of each denominator with bounded digits, as well as the distribution of those with minimal possible multiplicity, remain incompletely understood (Cohen, 2016).
• Non-Archimedean, Higher-Rank, and Function Field Analogues: Generalizations to continued fractions in non-real settings are under development, with partial analogues established in Hurwitz and $p$-adic settings (Robert et al., 2023, Lee, 12 Dec 2025).

Current approaches suggest that a combination of analytic, combinatorial, and spectral methods—possibly together with new insight into the algebraic and dynamical structure of continued fraction semigroups—will be needed to resolve the outstanding cases of Zaremba's conjecture.

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Key References:

• (Shulga, 2023) Radical bound for Zaremba’s conjecture
• (Huang, 2013) An Improvement To Zaremba’s Conjecture
• (Kan, 2014, Kan, 2015, Kan, 2016) Further density/proportion results
• (Robert et al., 2023) Complex Hurwitz analogue
• (Moshchevitin et al., 2022) Improved analytic bound for all $q$
• (Lee, 12 Dec 2025) Asymptotic statistics and complex analogues
• (Dubno, 2023) Algorithmic advances in explicit constructions
• (Cohen, 2016) Multiplicity and distribution heuristics
• (Coons, 2017) Mahler regularity interpretation

These works reflect the multifaceted advances in the paper of Zaremba’s conjecture, from explicit constructions to statistical density theorems, and their structural extensions across arithmetic and geometric settings.