One-dimensional Fibonacci quasilattices and their application to the Euclidean algorithm and Diophantine equations

The one-dimensional $\mathcal{L}$ quasilattices $\mathcal{F}^2=\mathcal{F}\times\mathcal{F}$ lying in the square Fibonacci quasilattice are classified; here $\mathcal{F}$ is the one-dimensional Fibonacci quasilattice. It is proved that there exists a countable set of similarity classes of quasilattices $\mathcal{L}$ in $\mathcal{F}^2$ (fine classification), and also four classes of local equivalence (rough classification).
Asymptotic distributions of points in quasilattices $\mathcal{L}$ are found and then applied to Diophantine equations involving the function $[\alpha]$ (the integral part of $\alpha$) and to equations of the form $A_1\circ X_1-A_2\circ X_2=C$ where the coefficients $C$ and $A_i$ and the variables take values in $\mathbb {N}=\{1,2,3,\dots\}$ and $\circ$ is Knuth's circular multiplication.