Beatty Sequences for a Quadratic Irrational: Decidability and Applications

Luke Schaeffer, Jeffrey Shallit, Stefan Zorcic

Let $\hat{I}\pm$ and ${\hat{I}}^2$ belong to the same quadratic field. We show that the inhomogeneous Beatty sequence $(\lfloor n\hat{I}\pm +{\hat{I}}^2\rfloor {)}_{n\ge 1}$ is synchronized, in the sense that there…