Continued Fractions and Gaps

Given a continued fraction, we construct a certain function that is discontinuous at every rational number p/q. We call this discontinuity the “gap”. We then try to characterize the gap sizes, and find, to the first order, the size is 1/q2, and that, for higher orders, the gap appears to be perfectly ’randomly’ distributed, in that it is Cauchy-dense on the unit square, and thus, this function has a fractal measure of exactly 2. We find this result to be very intriguing, as we know of no other functions that have this property (There are many fractal curves that have this property, but not functions. That is, a space-filling curve can be used to enumerate R2by R but such space-filling curves have locality properties induced by R that the gap function appears not to have). When examining this function for small rationals, some very curious algebraic relationships appear to relate various rationals. This paper is part of a set of chapters that explore the relationship between the real numbers, the modular group, and fractals. 1 Continued Fractions and Gaps 1.1 Definitions; Notation Given a real number 0 ≤ x ≤ 1, a sequence of integers [a1,a2, ...] can be found that define its continued-fraction expansion: x = 1/(a1 +1/(a2+1/(a3+ ...))) (1) Given any particular x, this sequence is straightforward to compute. Rational numbers x always have a finite number of terms in the sequence. For a rational number x, there are two distinct expansions that yield the same value: x+ = [a1,a2, ...,aN ] (2) and x− = [a1,a2, ...,aN −1,1] (3)