One-dimensional Fibonacci Quasilattices and Their Application to the Euclidean Algorithm and Diophantine Equations

The one-dimensional quasilattices L lying in the square Fibonacci quasilattice F2 = F×F are classified; here F is the one-dimensional Fibonacci quasilattice. It is proved that there exists a countable set of similarity classes of quasilattices L in F2 (fine classification), and also four classes of local equivalence (rough classification). Asymptotic distributions of points in quasilattices L are found and then applied to Diophantine equations involving the function [α] (the integral part of α) and to equations of the form A1 ◦X1−A2 ◦X2 = C, where the coefficients C and Ai and the variables Xi take values in N = {1, 2, 3, . . .} and ◦ is Knuth’s circular multiplication.