Beatty Sequences for a Quadratic Irrational: Decidability and Applications

Abstract

Let $α$ and $β$ belong to the same quadratic field. We show that the inhomogeneous Beatty sequence $(\lfloor n α+ β\rfloor)_{n \geq 1}$ is synchronized, in the sense that there is a finite automaton that takes as input the Ostrowski representations of $n$ and $y$ in parallel, and accepts if and only if $y = \lfloor n α+ β\rfloor$. Since it is already known that the addition relation is computable for Ostrowski representations based on a quadratic number, a consequence is a new and rather simple proof that the first-order logical theory of these sequences with addition is decidable. The decision procedure is easily implemented in the free software Walnut. As an application, we show that for each $r \geq 1$ it is decidable whether the set $\{ \lfloor n α+ β\rfloor \, : \, n \geq 1 \}$ forms an additive basis (or asymptotic additive basis) of order $r$. Using our techniques, we also solve some open problems of Reble and Kimberling, and give an explicit characterization of a sequence of Hildebrand et al.

Figures (4)

• Figure 1: Shift automaton.
• Figure 2: Synchronized automaton for $(\lfloor n\varphi + {1\over 2} \rfloor)_{n \geq 0}$.
• Figure 3: Fibonacci automaton for $c(n) - \tilde{c}(n)$. A state labeled $q/a$ has output $a$.
• Figure 4: Shift automaton.

Theorems & Definitions (32)

• Proposition 1
• Theorem 2
• Lemma 3
• proof
• Theorem 4
• proof
• Corollary 5
• proof
• Remark 6
• Theorem 7