McMullen's $M_d = 2$ question: infinitely many periodic continued fractions with partial quotients 1 and 2 in every real quadratic field

For every squarefree $d>1$, the real quadratic field $\mathbb{Q}(\sqrt{d})$ contains infinitely many purely periodic continued fractions with partial quotients bounded by some constant $M_d$ depending on the field — a 1980 theorem of Wilson that McMullen re-proved and generalized in 2009, where he asked: "In Theorem 1.1, can one take $M_d = 2$ for all $d$? That is, does every real quadratic field contain infinitely many periodic continued fractions with $1 \le a_i \le 2$?" The problem is open. Infinitely many individual quadratic fields are known to admit such expansions (for example $\mathbb{Q}(\sqrt{10})$, by Mercat), and the existence of a single purely periodic expansion with digits in $\{1,2\}$ has been verified by computer for every squarefree radicand below 127, but no result covers all real quadratic fields with any absolute digit bound in place of 2: as of 2018, Bourgain and Kontorovich recorded that even the much weaker statement that every field contains at least one element with some absolute digit bound was open. The natural local-global approach through thin semigroups was disproved in general by Rickards and Stange in 2025 via reciprocity obstructions, and recent progress on Zaremba's conjecture — including a 2026 preprint claiming the full conjecture with a large absolute bound — would at best yield large uniform bounds, not the bound 2. This record states the question in its source-faithful form, its equivalent geometric formulation via low-lying closed geodesics on the modular surface, and the known partial results.

Background

Every irrational real number $x$ has a unique simple continued fraction expansion $x = [a_0, a_1, a_2, \ldots] = a_0 + 1/(a_1 + 1/(a_2 + \cdots))$, where $a_0$ is an integer and the partial quotients $a_1, a_2, \ldots$ are positive integers. Lagrange proved that $x$ is a quadratic irrational (irrational of degree 2 over the rationals) if and only if its partial quotients are eventually periodic; Galois proved that the expansion is purely periodic — $a_{i+p} = a_i$ for some period $p$ and all $i \ge 0$, written $x = [a_0, \ldots, a_{p-1}]$ with an overbar on the repeating block — if and only if $x > 1$ and its algebraic conjugate lies in $(-1,0)$. Numbers with bounded partial quotients are exactly the badly approximable numbers, those satisfying $|x - p/q| > c/q^2$ for some $c>0$ and all rationals $p/q$; they form a set of Lebesgue measure zero but full Hausdorff dimension. The source paper refers to elements of a fixed field with a uniform bound on their partial quotients as uniformly Diophantine.

A remarkable converse to Lagrange's theorem holds: any fixed real quadratic field $\mathbb{Q}(\sqrt{d})$ contains infinitely many periodic continued fractions with uniformly bounded partial quotients. Woods (1978), in the language of the Markoff spectrum of an algebraic number field (the set of approximation constants attained by binary quadratic forms over the field), and Wilson (1980), in the language of the Lagrange spectrum of a quadratic field (the set of best rational-approximation constants of its elements), constructed periodic patterns with bounded digits in prescribed real quadratic fields; McMullen (2009) gave a new proof and embedded the result in a broader framework. His Theorem 1.1 asserts that any real quadratic field $\mathbb{Q}(\sqrt{d})$ contains infinitely many periodic continued fractions $x = [a_0, \ldots, a_{p-1}]$ (purely periodic, in the bar notation above) with $1 \le a_i \le M_d$, where the constant $M_d$ depends only on $d$; for instance $\mathbb{Q}(\sqrt{5})$ contains the infinite family $[\overline{1,4,2,3}]$, $[\overline{1,1,4,2,1,3}]$, $[\overline{1,1,1,4,2,1,1,3}], \ldots$, so $M_5 = 4$.

The theorem has two equivalent geometric reformulations. First, through the modular surface $\mathbb{H}/\mathrm{SL}_2(\mathbb{Z})$, the quotient of the hyperbolic upper half-plane by the group of $2 \times 2$ integer matrices of determinant 1: closed geodesics on this orbifold (a space that is a manifold away from the finitely many points with nontrivial group stabilizer) correspond, up to conjugacy, to ideal classes in orders of real quadratic fields, or equivalently to their purely periodic continued fractions, and a uniform bound on partial quotients is the same as the geodesic remaining in a fixed compact set. Theorem 1.2 asserts that for any fundamental closed geodesic $\gamma$ (one that is not a multiple of a strictly shorter one), some compact set contains infinitely many primitive closed geodesics (indivisible ones, not iterates of a shorter closed geodesic) whose lengths are integral multiples of $L(\gamma)$; the geodesics with lengths a multiple of $L(\gamma)$ are precisely those defined over the same real quadratic field. Second, through the geometry of numbers: an ideal $I$ in an order (a subring of the field that is a free $\mathbb{Z}$-module of full rank) embeds as a planar lattice under the two real embeddings, and its packing density $\delta(I)$ is the fraction of the plane covered by optimally placed equal disks centered at the lattice points; Theorem 1.4 asserts that any real quadratic field has infinitely many ideal classes, across its orders, with $\delta(I) > \delta_K > 0$. A smaller digit bound in Theorem 1.1 corresponds to denser packings and to geodesics confined to a smaller compact set.

The question is entangled with Zaremba's conjecture (1972, from quasi-Monte-Carlo numerical integration): there is an absolute constant $N$ such that every positive integer $f$ is the denominator of a rational $a/f$ whose finite continued fraction has all partial quotients at most $N$. The conjecture is expected to hold with $N = 5$, and Hensley conjectured that $N = 2$ should suffice after excluding finitely many denominators; it is known for denominators that are powers of 2 or 3 with $N = 3$. Bourgain and Kontorovich proved in 2014 that a density-one set of denominators works with $N = 50$ (Huang improved the bound to 5), and a 2026 preprint of Xin Zhang claims the full conjecture with a large absolute bound. Mercat proved that a Zaremba-type statement with constant $m$ implies that every real quadratic field contains infinitely many periodic continued fractions with digits at most $m+1$: the rational-side and field-side problems are cousins, the rational side being comparatively accessible.

Finally, the modern dynamical viewpoint: the set of reals with all partial quotients in a finite alphabet $A$ is a Cantor set, the limit set of a thin semigroup $\Gamma_A$ of $\mathrm{SL}_2(\mathbb{Z})$ (generated by the continued-fraction digit transformations, taken in pairs to keep determinant one); "thin" refers to such semigroups being sparse inside the ambient arithmetic group. Bourgain and Kontorovich proposed a local-global principle for these semigroups — when the limit set has Hausdorff dimension greater than $1/2$, every admissible integer (one not already excluded by congruence conditions) should arise as a trace or denominator — which, applied to the alphabet $\{1,2\}$, would imply an affirmative answer to the question below. Rickards and Stange disproved this principle in general in 2025 by exhibiting reciprocity obstructions: infinite families of missed values (for example, expansions $[0; a_1, \ldots, a_n, 1, 1, 2]$ with $a_i \in \{4, 8, \ldots, 128\}$ never have square denominators) invisible to congruence considerations. The alphabet $\{1,2\}$ and the original Zaremba conjecture are not among the refuted cases, but the flagship mechanism for such problems is now known to fail in general.

Problem Statement

The question is posed by McMullen (2009, Section 6, p. 842) as an explicit question about his Theorem 1.1, immediately after his discussion of Zaremba's conjecture:

"Question. In Theorem 1.1, can one take $M_d = 2$ for all $d$? That is, does every real quadratic field contain infinitely many periodic continued fractions with $1 \le a_i \le 2$?"

Here Theorem 1.1 asserts that any real quadratic field $\mathbb{Q}(\sqrt{d})$ contains infinitely many periodic continued fractions $x = [a_0, \ldots, a_{p-1}]$ with $1 \le a_i \le M_d$, where the paper's bar notation means the expansion is purely periodic, $a_{i+p} = a_i$ for all $i \ge 0$. Unpacked, the question asks:

Does every real quadratic field $K = \mathbb{Q}(\sqrt{d})$, equivalently every squarefree integer $d > 1$, contain infinitely many distinct purely periodic continued fractions with all partial quotients in the two-letter alphabet $\{1, 2\}$?

A purely periodic expansion using only the digit 1 is necessarily the golden ratio $[\overline{1}] = (1+\sqrt{5})/2$, which lies in the single field $\mathbb{Q}(\sqrt{5})$, so $\{1,2\}$ is the smallest nontrivial bounded alphabet and the question asks for the optimal uniform version of Theorem 1.1. An affirmative answer must hold for every real quadratic field with the digit bound exactly 2; it does not suffice to establish any absolute bound larger than 2, to treat a subfamily of fields (for example fields of small regulator, the regulator being the logarithm of the fundamental unit, that is, the smallest unit of the field greater than 1), or to produce elements that are merely eventually periodic with digits in $\{1,2\}$ (prepending digits to one purely periodic $\{1,2\}$ expansion in a field always yields more eventually periodic expansions in the same field, so the eventual-periodic version is strictly weaker). A negative answer would consist of one explicit real quadratic field together with a proof that it contains only finitely many purely periodic continued fractions with all partial quotients in $\{1,2\}$. By the equivalences established in the source, an affirmative answer is equivalent to: for every fundamental closed geodesic $\gamma$ on the modular surface, the compact set corresponding to the digit bound 2 contains infinitely many primitive closed geodesics whose lengths are integral multiples of $L(\gamma)$; and it implies that in every real quadratic field, infinitely many ideal classes of orders have packing density bounded below by an explicit positive constant depending only on the digit bound 2. The expected answer is affirmative, in line with Hensley's conjecture that even the rational-side $\{1,2\}$ statement should hold up to finitely many exceptions.

References

1. Curtis T. McMullen, Uniformly Diophantine numbers in a fixed real quadratic field, Compositio Mathematica 145 (2009), no. 4, 827-844, DOI 10.1112/S0010437X09004102, https://doi.org/10.1112/S0010437X09004102 (the primary source).
2. A. C. Woods, The Markoff spectrum of an algebraic number field, Journal of the Australian Mathematical Society (Series A) 25 (1978), no. 4, 486-488, DOI 10.1017/S1446788700021467, https://doi.org/10.1017/S1446788700021467.
3. S. M. J. Wilson, Limit points in the Lagrange spectrum of a quadratic field, Bulletin de la Société Mathématique de France 108 (1980), 137-141, DOI 10.24033/bsmf.1913, https://doi.org/10.24033/bsmf.1913.
4. Douglas Hensley, A polynomial time algorithm for the Hausdorff dimension of continued fraction Cantor sets, Journal of Number Theory 58 (1996), no. 1, 9-45, DOI 10.1006/jnth.1996.0058, https://doi.org/10.1006/jnth.1996.0058 (his $N=2$ conjecture is cited from McMullen 2009, Section 6).
5. Paul Mercat, Construction de fractions continues périodiques uniformément bornées, Journal de Théorie des Nombres de Bordeaux 25 (2013), no. 1, 111-146, DOI 10.5802/jtnb.829, https://doi.org/10.5802/jtnb.829.
6. Paul Mercat, Construction of uniformly bounded periodic continued fractions, arXiv:1601.08109 (2016), https://arxiv.org/abs/1601.08109 (expanded preprint).
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