For N ∈ N_≥2 and α ∈ R such that 0 < α ≤ N − 1, the continued fraction map T_α: [α, α+1] → [α, α+1) is defined as T_α (x):= N/x−d(x), where d: [α, α+1] → N is defined by d(x):= ⌊N/x − α⌋. A maximal open interval (a, b) ⊂ I_α is called a gap of I_α if for almost every x ∈ I_α there is an n₀ (x) ∈ N such that x_n /∈ (a, b) for all n ≥ n₀ . In this paper, all conditions are given in which I_α is gapless. For α =^√N − 1 it is shown that the number of gaps is a finite, monotonically non-decreasing and unbounded function of N.

@en

For N∈ N_{≥ 2} and α∈ R such that 0<α≤N-1, we define I_α: = [α, α+ 1] and Iα-:=[α,α+1) and investigate the continued fraction map Tα:Iα→Iα-, which is defined as Tα(x):=Nx-d(x), where d: I_α→ N is defined by d(x):=⌊Nx-α⌋. For N∈ N_{≥ 7}, for certain values of α, open intervals (a, b) ⊂ I_α exist such that for almost every x∈ I_α there is an n∈ N for which Tαn(x)∉(a,b) for all n≥ n. These gaps (a, b) are investigated using the square Υα:=Iα×Iα-, where the orbitsTαk(x),k=0,1,2,… of numbers x∈ I_α are represented as cobwebs. The squares Υ_α are the union of fundamental regions, which are related to the cylinder sets of the map T_α, according to the finitely many values of d in T_α. In this paper some clear conditions are found under which I_α is gapless. If I_α consists of at least five cylinder sets, it is always gapless. In the case of four cylinder sets there are usually no gaps, except for the rare cases that there is one, very wide gap. Gaplessness in the case of two or three cylinder sets depends on the position of the endpoints of I_α with regard to the fixed points of I_α under T_α.

@en

In this thesis continued fractions are studied in three directions: semi-regular continued fractions, Nakada’s α-expansions and N-expansions. In Chapter 1 the general concept of a continued fraction is given, involving an operator that yields the partial quotients or digits of a continued fraction expansion. The approximation coefficients θ_n(x) := q²|x-p_n/q_n| are introduced, where p_n/q_n, n ∈ 0, 1, 2, . . ., are the convergents of the continued fraction. Some well-known results on semi-regular continued fractions are given. Finally, the concept of ‘natural extension’ is explained. Chapter 2 is about orders (called patterns) of triplets of three consecutive approximation coefficients θ_(n-1)(x), θ_n(x) and θ_(n+1)(x). The asymptotic frequency of pattern Χ(n) is defined by AF(X(n)) := lim_(N→∞) 1/N #{n ∈ N | 2 ≤ n ≤ N, X(n)}. Starting with the regular continued fraction (RCF), it is shown that, for instance, the asymptotic frequency as n → ∞of the pattern θ_(n-1)(x) < θ_n(x) < θ_(n+1)(x) is smaller than the asymptotic frequency of the pattern θ_n(x) < θ_(n+1)(x) < θ_(n-1)(x). The asymptotic frequencies in the case of the RCF are explicitly given: two of them are 0.1210..., the others are 0.1894... . After this, these patterns are studied of two other semi-regular continued fractions: the optimal continued fraction (OCF) and the nearest integer continued fraction (NICF). The asymptotic frequencies of the OCF prove to be more equally distributed: the two less frequent patterns of the RCF now have the asymptotic frequency 0.1603... , where this is 0.1698... for the other patterns. The asymptotic frequencies of the NICF prove to be different for all six patterns. However, summation of specific pairs yield once 2 · 0.1603... and two times 2 · 0.1698... , thus showing a great correspondence with the OCF. Chapter 3 is dedicated to the natural extension of Nakada’s α-expansions. By meansof singularisations and insertions in these continued fraction expansions, involving the removal or addition of partial quotients 1 in exchange with partial quotients with a minus sign, the interval on which the natural extension of Nakada’s continued fractionmap T_α is given is extended from [√2-1,1) to [(√10-3)/2,1). From our construction it followsthat ­Ω_α, the domain of the natural extension of T_α, is metrically isomorphic to ­g for α ∈ [g², g), where g is the small golden mean. Finally, although ­Ω_α proves to be very intricate and unmanageable for α ∈ [g², (√10-3)/2), the α-Legendre constant L(α) on this interval is explicitly given. In Chapter 4 N-expansions are introduced for natural numbers N larger than 1. These expansions, like semi-regular continued fraction expansions, are also sequences of partial quotients, called orbits, existing in the interval I_α = [α,α+1] for some α ∈ (0,√N-1]. Depending on N and α, there is a finite number of consecutive digits that occur as partial quotient. It appears that there are conditions (that is, combinations of N and α) such that these orbits eventually do not land in certain parts of the interval I_α, called gaps. It is proved that if the number of digits is at least five, no gaps exist. If the number of digits is four, there do not exist gaps for most N, but in the cases that there are α such that I_α contains a gap, there is only one and it covers the lion’s part of I_α. When the number of digits is two or three, the number of gaps varies, but it is possible to give very clear conditions under which there are no gaps.@en

Denote by p n /q n ,n=1,2,3,…, pn/qn,n=1,2,3,…, the sequence of continued fraction convergents of a real irrational number x x . Define the sequence of approximation coefficients by θ n (x):=q n |q n x−p n |,n=1,2,3,… θn(x):=qn|qnx−pn|,n=1,2,3,… . In the case of regular continued fractions the six possible patterns of three consecutive approximation coefficients, such as θ n−1 <θ n <θ n+1  θn−1<θn<θn+1 , occur for almost all x 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.@en

By means of singularisations and insertions in Nakada's α-expansions, which involves the removal of partial quotients 1 while introducing partial quotients with a minus sign, the natural extension of Nakada's continued fraction map Tα is given for (10-2)/3≤α<1. From our construction it follows that Ωα, the domain of the natural extension of Tα, is metrically isomorphic to Ωg for α∈[g2,g), where g is the small golden mean. Finally, although Ωα proves to be very intricate and unmanageable for α∈[g2,(10-2)/3), the α-Legendre constant L(α) on this interval is explicitly given.

@en