Gaps, Frequencies and Spacial Limits of Continued Fraction Expansions

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.