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

Abstract:

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.