Self-Matching Properties of Beatty Sequences

Abstract

We study the selfmatching properties of Beatty sequences, in particular of the graph of the function $\lfloor jβ\rfloor $ against $j$ for every quadratic unit $β\in(0,1)$. We show that translation in the argument by an element $G_i$ of generalized Fibonacci sequence causes almost always the translation of the value of function by $G_{i-1}$. More precisely, for fixed $i\in\N$, we have $\bigl\lfloor β(j+G_i)\bigr\rfloor = \lfloor βj\rfloor +G_{i-1}$, where $j\notin U_i$. We determine the set $U_i$ of mismatches and show that it has a low frequency, namely $β^i$.