An algorithm for constructing multidimensional continued fractions and linear dependence of numbers

Based on work done by Morse and Hedlund (1940) it was observed by Arnoux and Rauzy (1991) that the classical continued fraction algorithm provides a surprising link between arithmetic and diophantine properties of an irrational number $\alpha$, the rotation by $\alpha$ on the torus $\mathbb{T} = \mathbb{R}/\mathbb{Z}$, and combinatorial properties of the well known Sturmian sequences, a class of sequences on two letters with low subword complexity. It has been conjectured since the early 1990ies that this correspondence carries over to generalized continued fraction algorithms, rotations on higher dimensional tori, and so-called $S$-adic sequences generated by substitutions. The idea of working towards this generalization is known as Rauzy’s program. Although, starting with Rauzy (1982) a number of examples for such a generalization was devised, Cassaigne, Ferenczi, and Zamboni (2000) came up with a counterexample that showed the limitations of such a generalization. Nevertheless, recently Berthe, Steiner, and Thuswaldner (2016) made some further progress on Rauzy’s program and were able to set up a generalization of the above correspondences. They proved that the above conjecture is true under certain natural conditions. A prominent role in this generalization is played by tilings induced by generalizations of the classical Rauzy fractal introduced by Rauzy (1982). Another idea which is related to the above results goes back to Artin (1924), who observed that the classical continued fraction algorithm and its natural extension can be viewed as a Poincare section of the geodesic flow on the space $SL_2(\mathbb{Z}) \ SL_2(\mathbb{R})$. Arnoux and Fisher (2001) revisited Artin’s idea and showed that the above mentioned correspondence between continued fractions, rotations, and Sturmian sequences can be interpreted in a very nice way in terms of an extension of this geodesic flow which they called the scenery flow. Currently, Arnoux et al. are setting up elements of a generalization of this connection as well. It is the aim of my series of lectures to review the above results.

Let x∈(0,1) be an irrational number with continued fraction expansion [a1(x),a2(x),⋯,an(x),⋯] . We give the multifractal spectrum of the irrationality exponent and the convergence exponent of x defined by v(x):=sup{v>0:|x−pq|<1qv for infinitely many (q,p)∈ℕ×ℤ} and τx:=infs⩾0:∑n⩾1an−sx<∞ respectively. To be precise, we completely determine the Hausdorff dimension of Eα,v=x∈0,1: τx=α, vx=v for any α⩾0, v⩾2 .

Appendix 1. MATLAB, Primes, Irrationals, and Continued Fractions

In his paper "Multidimensional continued fractions" ( Ann. Univ. Sci. Budapest. Eötvös Sect. Math. , v. 13, 1970, pp. 113-140), G. Szekeres introduced a new higher dimensional analogue of the ordinary continued fraction expansion of a single real number. The Szekeres algorithm associates with each k -tuple ( α 1 , … , α k ) ({\alpha _1}, \ldots ,{\alpha _k}) of real numbers (satisfying 0 &gt; α i &gt; 1 0 &gt; {\alpha _i} &gt; 1 ) a sequence b 1 , b 2 , … {b_1}, {b_2}, \ldots of positive integers; this sequence is called a continued k -fraction, and for k = 1 it is just the sequence of partial quotients of the ordinary continued fraction for α 1 {\alpha _1} . A simple recursive procedure applied to b 1 , b 2 , … {b_1}, {b_2}, \ldots produces a sequence a ( n ) = ( A n ( 1 ) / B n , … , A n ( k ) / B n ) ( n = 1 , 2 , … ; A n ( i ) ⩾ 0 a(n) = (A_n^{(1)}/{B_n}, \ldots ,A_n^{(k)}/{B_n})\;(n = 1,2, \ldots ;A_n^{(i)} \geqslant 0 and B n &gt; 0 {B_n} &gt; 0 are integers) of simultaneous rational approximations to ( α 1 , … , α k ) ({\alpha _1}, \ldots ,{\alpha _k}) and a sequence c ( n ) = ( c n 0 , c n 1 , … , c n k ) ( n = 1 , 2 , … ) c(n) = ({c_{n0}},{c_{n1}}, \ldots ,{c_{nk}})\;(n = 1,2, \ldots ) of integer ( k + 1 ) (k + 1) -tuples such that the linear combination c n 0 + c n 1 α 1 + ⋯ + c n k α k {c_{n0}} + {c_{n1}}{\alpha _1} + \cdots + {c_{nk}}{\alpha _k} approximates zero. Szekeres conjectured, on the basis of extensive computations, that the sequence a ( 1 ) , a ( 2 ) , … a(1),a(2), \ldots contains all of the "best" simultaneous rational approximations to ( α 1 , … , α k ) ({\alpha _1}, \ldots ,{\alpha _k}) and that the sequence c ( 1 ) , c ( 2 ) , … c(1),c(2), \ldots contains all of the "best" approximations to zero by the linear form x 0 + x 1 α 1 + ⋯ + x n α n {x_0} + {x_1}{\alpha _1} + \cdots + {x_n}{\alpha _n} . For the special case k = 2 and α 1 = θ 2 − 1 , α 2 = θ − 1 {\alpha _1} = {\theta ^2} - 1,{\alpha _2} = \theta - 1 (where θ = 2 cos ⁡ ( 2 π / 7 ) \theta = 2\cos (2\pi /7) is the positive root of x 3 + x 2 − 2 x − 1 = 0 {x^3} + {x^2} - 2x - 1 = 0 , Szekeres further conjectured that the 2-fraction b 1 , b 2 , … {b_1},{b_2}, \ldots is "almost periodic" in a precisely defined sense. In this paper the Szekeres conjectures concerning best approximations to zero by the linear form x 0 + x 1 ( θ 2 − 1 ) + x 2 ( θ − 1 ) {x_0} + {x_1}({\theta ^2} - 1) + {x_2}(\theta - 1) and concerning almost periodicity for the 2-fraction of ( θ 2 − 1 , θ − 1 ) ({\theta ^2} - 1,\theta - 1) are proved. The method used can be applied to other pairs of cubic irrationals α 1 , α 2 {\alpha _1},{\alpha _2} .

Optimal continued fractions

Let $\beta > 1$ be a real number and $x \in [0,1)$ be an irrational number. We denote by $k_n(x)$ the exact number of partial quotients in the continued fraction expansion of $x$ given by the first $n$ digits in the $\beta$-expansion of $x$ ($n \in \mathbb{N}$). It is known that $k_n(x)/n$ converges to $(6\log2\log\beta)/\pi^2$ almost everywhere in the sense of Lebesgue measure. In this paper, we improve this result by proving that the Lebesgue measure of the set of $x \in [0,1)$ for which $k_n(x)/n$ deviates away from $(6\log2\log\beta)/\pi^2$ decays to 0 exponentially as $n$ tends to $\infty$, which generalizes the result of Faivre \cite{lesFai97} from $\beta = 10$ to any $\beta >1$. Moreover, we also discuss which of the $\beta$-expansion and continued fraction expansion yields the better approximations of real numbers.

Khinchin proved that the arithmetic mean of the regular continued fraction digits of Lebesgue almost every irrational number in (0, 1) diverges to infinity. Hence, none of the classical limit theorems such as the weak and strong laws of large numbers or central limit theorems hold. Nevertheless, we prove the existence of a large deviations rate function which estimates exponential probabilities with which the arithmetic mean of digits stays away from infinity. This leads us to a contradiction to the widely-shared view that the large deviation principle is a refinement of laws of large numbers: the former can be more universal than the latter.

Let α be an irrational number with simple continued fraction α = (a0, a1, a2,…). The problem studied is that of whether the sequence (qn) of denominators of the convergents pn/qn to α has a subsequence (Bn) = (qin} which is the sequence of denominators of convergents An/Bn to a different number α′. In other words, does there exist a subsequence {qin} which satisfies qio= 1 and For example, the sequence of denominators of convergents to ½(3 − e) is a subsequence of the sequence of denominators of convergents to ε.

Denote by $p_n/q_n, n=1,2,3,\ldots,$ the sequence of continued fraction convergents of a real irrational number $x$. Define the sequence of approximation coefficients by $\theta_n(x):=q_n\left|q_nx-p_n\right|, n=1,2,3,\ldots$. In the case of regular continued fractions the six possible patterns of three consecutive approximation coefficients, such as $\theta_{n-1}<\theta_n<\theta_{n+1}$, occur for almost all $x$ with only two different asymptotic frequencies. In this paper it is shown how these asymptotic frequencies can be determined for two other semi-regular cases. It appears that the optimal continued fraction has a similar distribution of only two asymptotic frequencies, albeit with different values. The six different values that are found in the case of the nearest integer continued fraction will show to be closely related to those of the optimal continued fraction.

Hankel continued fraction and its applications

This paper extends, in a multi-dimensional framework, pattern recognition technics for generation or recognition of digital lines. More precisely, we show how the connection between chain codes of digital lines and continued fractions can be generalized by a connection between tilings and multi-dimensional continued fractions. This leads to a new approach for generating and recognizing digital hyperplanes.

Let ξ be an irrational number with simple continued fraction expansion ξ = [a0;a1,a2,…,an,…], let pn/qn be its nth convergent and let θn = qn|qnξ − pn|. In this paper a general method is introduced to deduce a series of inequalities involving the triple (θn−1, θn, θn+1).

It has been a long-standing problem to find good symbolic codings for translations on the d -dimensional torus that enjoy the beautiful properties of Sturmian sequences like low factor complexity and good local discrepancy properties. Inspired by Rauzy’s approach we construct such codings in terms of multidimensional continued fraction algorithms that are realized by sequences of substitutions. In particular, given any exponentially convergent continued fraction algorithm, these sequences lead to renormalization schemes which produce symbolic codings of toral translations and bounded remainder sets at all scales in a natural way. The exponential convergence properties of a continued fraction algorithm can be viewed in terms of a Pisot type condition imposed on an attached symbolic dynamical system. Using this fact, our approach provides a systematic way to confirm purely discrete spectrum results for wide classes of symbolic dynamical systems. Indeed, as our examples illustrate, we are able to confirm the Pisot conjecture for many well-known families of sequences of substitutions. These examples include classical algorithms like the Jacobi–Perron, Brun, Cassaigne–Selmer, and Arnoux–Rauzy algorithms. As a consequence, we gain symbolic codings of almost all translations of the 2-dimensional torus having factor complexity 2n + 1 that are balanced for words, which leads to multiscale bounded remainder sets. Using the Brun algorithm, we also give symbolic codings of almost all 3-dimensional toral translations having multiscale bounded remainder sets.

A simplex–karyon algorithm for expanding real numbers α = (α1,..., α d ) in multidimensional continued fractions is considered. The algorithm is based on a (d + 1)-dimensional superspace S with embedded hyperplanes: a karyon hyperplane K and a Farey hyperplane F. The approximation of numbers α by continued fractions is performed on the hyperplane F, and the degree of approximation is controlled on the hyperplane K. A local ℘(r)-strategy for constructing convergents is chosen, with a free objective function ℘(r) on the hyperplane K.

By a result of Hardy-Littlewood concerning the growth of the sums \(\sum\nolimits_{n = 1}^N {{e^{i\pi {n^2}x}}} \), for each s ∈ (1/2, 1], the series \({F_s}(x) = \sum\nolimits_{n = 1}^\infty {{e^{i\pi {n^2}x/{n^s}}}} \) converges almost everywhere but not everywhere on [−1, 1]. However, there does not yet exist an intrinsic description of the set of convergence for Fs. In this paper, we define in terms of even continued fractions a subset of [−1, 1] of full measure where the series converges. As an intermediate step, we prove that for s > 0, the sequence of functions $$\sum\limits_{n = 1}^N {\frac{{{e^{i\pi {n^2}x}}}}{{{n^S}}} - {e^{{\text{sign}}(x)i\frac{\pi }{4}}}|x{|^{S - \frac{1}{2}}}\sum\limits_{n = 1}^{\left\lfloor {N|x|} \right\rfloor } {\frac{{{e^{ - i\pi {n^2}/x}}}}{{{n^S}}}} } $$ converges as N → ∞ to a function Ωs continuous on [−1, 1] {0} with (at most) a singularity at x = 0 of type x(s−1)/2 (s ≠ 1) or a logarithmic singularity (s = 1). We provide an explicit expression for Ωs and the error term. Finally, we study thoroughly the convergence properties of certain series defined in terms of the convergents of the even continued fraction of an irrational number.