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

The one-dimensional quasilattices $\mathcal{L}$ lying in the square Fibonacci quasilattice $\mathcal{F}^2=\mathcal{F} \times \mathcal{F}$ 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 $X_i$ take values in $\mathbb{N}=\lbrace 1,2,3,\ldots \rbrace$ and $\circ$ is Knuth's circular multiplication.