Hausdorff dimension of sets of continued fractions with unbounded partial quotients along subsequence

Numbers whose continued fraction expansion contains only small digits have been extensively studied. In the real case, the Hausdorff dimension σ M of the reals with digits in their continued fraction expansion bounded by M was considered, and estimates of σ M for M→∞ were provided by Hensley (J. Number Theory 40:336–358, 1992). In the rational case, first studies by Cusick (Mathematika 24:166–172, 1997), Hensley (In: Proc. Int. Conference on Number Theory, Quebec, pp. 371–385, 1987) and Vallée (J. Number Theory 72:183–235, 1998) considered the case of a fixed bound M when the denominator N tends to ∞. Later, Hensley (Pac. J. Math. 151(2):237–255, 1991) dealt with the case of a bound M which may depend on the denominator N, and obtained a precise estimate on the cardinality of rational numbers of denominator less than N whose digits (in the continued fraction expansion) are less than M(N), provided the bound M(N) is large enough with respect to N. This paper improves this last result of Hensley towards four directions. First, it considers various continued fraction expansions; second, it deals with various probability settings (and not only the uniform probability); third, it studies the case of all possible sequences M(N), with the only restriction that M(N) is at least equal to a given constant M 0; fourth, it refines the estimates due to Hensley, in the cases that are studied by Hensley. This paper also generalises previous estimates due to Hensley (J. Number Theory 40:336–358, 1992) about the Hausdorff dimension σ M to the case of other continued fraction expansions. The method used in the paper combines techniques from analytic combinatorics and dynamical systems and it is an instance of the Dynamical Analysis paradigm introduced by Vallée (J. Théor. Nr. Bordx. 12:531–570, 2000), and refined by Baladi and Vallée (J. Number Theory 110:331–386, 2005).

For regular continued fraction, if a real number [Formula: see text] and its rational approximation [Formula: see text] satisfying [Formula: see text], then, after deleting the last integer of the partial quotients of [Formula: see text], the sequence of the remaining partial quotients is a prefix of that of [Formula: see text]. In this paper, we show that the situation is completely different if we consider the Hurwitz continued fraction expansions of a complex number and its rational approximations. More specifically, we consider the set [Formula: see text] of complex numbers which are well approximated with the given bound [Formula: see text] and have quite different Hurwitz continued fraction expansions from that of their rational approximations. The Hausdorff and packing dimensions of such set are determined. It turns out that its packing dimension is always full for any given approximation bound [Formula: see text] and its Hausdorff dimension is equal to that of the [Formula: see text]-approximable set [Formula: see text] of complex numbers. As a consequence, we also obtain an analogue of the classical Jarník theorem in real case.

A fundamental challenge within the metric theory of continued fractions involves quantifying sets of real numbers especially when their partial quotients exhibit specific growth rates. For any positive function Φ, the Wang–Wu theorem (2008) comprehensively describes the Hausdorff dimension of the set E 1 ( Φ ) : = { x ∈ [ 0 , 1 ) : a n ( x ) ≥ Φ ( n ) for infinitely many n ∈ N } . \mathcal{E}_{1}(\Phi):=\{x\in[0,1):a_{n}(x)\geq\Phi(n)\ \text{for infinitely many}\ n\in\mathbb{N}\}. Various generalisations of this set exist, such as substituting one partial quotient with the product of consecutive partial quotients in the aforementioned set which has connections with the improvements to Dirichlet’s theorem, and many other sets of similar nature. Establishing the upper bound of the Hausdorff dimension of such sets is significantly easier than proving the lower bound. In this paper, we present a unified approach to get an optimal lower bound for many known setups, including results by Wang–Wu [Adv. Math. (2008)], Huang–Wu–Xu [Israel J. Math. (2020)], Tan–Zhou [Nonlinearity (2023)], and several others. We also provide a new theorem derived as an application of our main result. We do this by finding an exact Hausdorff dimension of the set S m ⁢ ( A 0 , … , A m − 1 ) ⁢ = def ⁢ { x ∈ [ 0 , 1 ) : c i ⁢ A i n ≤ a n + i ⁢ ( x ) < 2 ⁢ c i ⁢ A i n , 0 ≤ i ≤ m − 1 , for infinitely many ⁢ n ∈ N } , S_{m}(A_{0},\ldots,A_{m-1})\overset{\mathrm{def}}{=}\{x\in[0,1):c_{i}A_{i}^{n}\leq a_{n+i}(x)<2c_{i}A_{i}^{n},\,0\leq i\leq m-1,\,\text{for infinitely many}\ n\in\mathbb{N}\}, where each partial quotient grows exponentially and the base is given by a parameter A i > 1 A_{i}>1 . For proper choices of A i A_{i} , this set serves as a subset for sets under consideration, providing an optimal lower bound of Hausdorff dimension in all of them. The crux of the proof lies in introducing multiple probability measures consistently distributed over the Cantor-type subset of S m ⁢ ( A 0 , … , A m − 1 ) S_{m}(A_{0},\ldots,A_{m-1}) .

The paper reports a link between the Hausdorff dimension of a number theoretically based set, and certain arithmetic properties of the spacing distribution of the two-dimensional harmonic oscillator. It is shown that the set of points , with continued fraction , such that diverges, has Hausdorff dimension . The set of convergents , such that the series diverge, is also shown to have a Hausdorff dimension . Although this result can be seen as a purely number-theoretic result, it is related to level spacing distributions in the following manner. For the two-dimensional harmonic oscillator with frequency ratio, , that has a continued fraction satisfying the above condition, the level spacing distribution is . Thus, the non-ergodic behaviour of the two-dimensional oscillator has Hausdorff dimension . Similar results are found for the system of a particle trapped in a box, using a number-theoretic result of Ramanujan.

We survey the dynamical systems side of the theory of continued fractions and touch on some of the frontiers of the subject. Ergodic theory plays a role. The work of Baladi and Vallée is discussed. Power series methods that allow for the computation of various numbers such as the Hausdorff dimensionof a continued fraction Cantor set, or the Wirsing constant of a particular continued fraction algorithm, to high accuracy, are also discussed.

Continued fraction Cantor sets, Hausdorff dimension, and functional analysis

In the one-dimensional Diophantine approximation, by using the continued fractions, Khintchine’s theorem and Jarnik’s theorem are concerned with the growth of the large partial quotients, while the improvability of Dirichlet’s theorem is concerned with the growth of the product of consecutive partial quotients. This paper aims to establish a complete characterization on the metric properties of the product of the partial quotients, including the Lebesgue measure-theoretic result and the Hausdorff dimensional result. More precisely, for any x ∈ [0, 1), let x =[a1, a2, …] beits continued fraction expansion. The size of the following set, in the sense of Lebesgue measure and Hausdorff dimension, Em(ϕ):= {x ∈ [0, 1): an (x) ⋯ an+m−1 (x) ≥ ϕ(n) for infinitely many n ∈ ℕ}, are given completely, where m ≥ 1 is an integer and ϕ: ℕ → ℝ+ is a positive function.

Good’s Theorem for regular continued fraction states that the set of real numbers [Formula: see text] such that [Formula: see text] has Hausdorff dimension [Formula: see text]. We show an analogous result for the complex plane and Hurwitz Continued Fractions: the set of complex numbers whose Hurwitz Continued fraction [Formula: see text] satisfies [Formula: see text] has Hausdorff dimension [Formula: see text], half of the ambient space’s dimension.

We establish that effective continued fraction dimension originally defined using s -gales [ 21 ] is robust, but surprisingly, that the effective continued fraction dimension and effective (base- b ) Hausdorff dimension of the same real can be unequal in general. We initially provide an equivalent characterization of continued fraction dimension using Kolmogorov complexity. We also prove new bounds on the Lebesgue measure of continued fraction cylinders, which may be of independent interest. We apply these bounds to reveal an unexpected behavior of continued fraction dimension. It is known that effective dimension is invariant with respect to base conversion [ 8 ]. We also know that Martin-Löf randomness and computable randomness are invariant not only with respect to base conversion, but also with respect to the continued fraction representation [ 21 ]. In contrast, for any \(0 \lt \varepsilon \lt 0.5\) , we prove the existence of a real whose effective Hausdorff dimension is less than \(\varepsilon\) but whose effective continued fraction dimension is greater than or equal to 0.5. This phenomenon is related to the “non-faithfulness” of certain families of covers [ 1 , 23 ]. We also establish that, for any real number, the effective continued fraction dimension of the real number is always greater than or equal to its effective Hausdorff dimension.

The study of products of consecutive partial quotients in the continued fraction arises naturally out of the improvements to Dirichlet’s theorem. We study the distribution of the two large products of partial quotients among the first n terms. More precisely, writing [a1(x),a2(x),…] the continued fraction expansion of an irrational number x∈(0,1) , for a non-decreasing function φ:N→R , we completely determine the size of the set F2(φ)=x∈[0,1):∃1⩽k≠l⩽n, ak(x)ak+1(x)⩾φ(n),al(x)al+1(x)⩾φ(n) for infinitely many n∈N in terms of Lebesgue measure and Hausdorff dimension.

A class of Cantor sets associated with the regular continued fractions

Porosity in Conformal Infinite Iterated Function Systems

The book gives an up to date overview of various aspects of multidimensional continued fractions, which are here defined through iteration of piecewise fractional linear maps. This includes the algorithms of Jacobi-Perron, Güting, Brun, and Selmer but it also includes continued fractions on simplices which are related to interval exchange maps or the Parry-Daniels map. New classes of subtractive algorithms are also included and the metric properties of these algorithms can be therefore investigated by methods of ergodic theory. The recent connection between multiplicative ergodic theory and Diophantine approximation presented, as well as several results on convergence and Perron's approach to periodicity, which has never appeared in book despite being published in 1907. Further chapters include the basic properties of continued fractions in the complex plane, connections with Hausdorff dimension and the Kuzmin theory for multidimensional maps.

Given x ∈ (0, 1), let [a1(x), a2(x), a3(x),…] be the continued fraction expansion of x and [Formula: see text] be the sequence of rational convergents. Good [The fractional dimensional theory of continued fractions, Math. Proc. Cambridge Philos. Soc.37 (1941) 199–228] discussed the growth properties of {an(x), n ≥ 1} and proved that for any β > 0, the set [Formula: see text] is of Hausdorff dimension [Formula: see text]. In this paper, we consider, for any β > 0, the set [Formula: see text] and show that the Hausdorff dimension of F(β) is [Formula: see text].

Rigorous effective bounds on the Hausdorff dimension of continued fraction Cantor sets: A hundred decimal digits for the dimension of E2