A rational number Rn is defined as In / Kn , where In, Kn are integer numbers. If the size of the integers In, Kn is limited to M, one obtains a finite number N of different rational numbers Rn, n=1...N. If the index n is defined in such a way that Rn-1<Rn (n-1 is an index!), one finds that there is a strange structure of the distances Dn=Rn-Rn-1 of neighbor rational numbers Rn and Rn-1 (for N toward infinity). Plots of this structure for N=100 are shown below.
Somehow similar structures are obtained form Farey series and from the Stern-Brocot tree - provided that I correctly understood the hints of Gareth McCaughan. Although the structure seems to be self-similar and looks like a fractal, its fractal dimension is 1.