[Paper Review] On the Finiteness property of negative cubic Pisot bases

Tomáš Vávra|arXiv (Cornell University)|Apr 4, 2014

Mathematical Dynamics and Fractals2 references3 citations

TL;DR

This paper proves the finiteness property (Property -F) for negative cubic Pisot bases $-\beta$, where $\beta$ is the real root of $x^3 - mx^2 - mx - m$ for $m \in \mathbb{N}$, and constructs an explicit finite state transducer for addition and subtraction in the negative Tribonacci base $-\gamma$. The transducer ensures bounded fractional digit growth, with a precise upper bound of 5 fractional digits for $(-\gamma)$-integers.

ABSTRACT

We study arithmetical aspects of Ito-Sadahiro number systems with negative base. We show that the bases $-β

Motivation & Objective

• To extend the class of negative non-integer bases known to satisfy the finiteness property (-F), which ensures that finite expansions form a ring.
• To provide a constructive algorithm for addition and subtraction in the negative Tribonacci base $-\gamma$, where $\gamma$ is the real root of $x^3 - x^2 - x - 1$.
• To determine the maximal number of fractional digits arising from addition or subtraction of two $(-\gamma)$-integers.
• To offer a concrete, explicit finite state transducer for arithmetic operations in $(-\gamma)$-numeration, overcoming the non-constructive nature of prior existence proofs.

Proposed method

• Uses Ito-Sadahiro's framework for $(-\beta)$-expansions with base $-\beta < -1$, defining expansions over the interval $[\ell, \ell+1)$ with $\ell = -\beta/(\beta+1)$.
• Applies the alternate lexicographic ordering condition to characterize admissible digit strings, ensuring uniqueness and correctness of expansions.
• Constructs a finite state transducer with 13 distinct states $Q_1$ to $Q_{13}$, each representing a specific configuration of digit windows and carry conditions.
• Defines transitions between states using rewriting rules based on digit patterns, such as $\overline{1}\overline{m}m\overline{n}$ and $1m\overline{m}n$, to simulate digit-wise addition with carry propagation.
• Verifies termination by showing that all paths eventually reach a state where no further rewriting is needed, due to the absence of infinite loops under the given constraints.
• Employs a systematic state classification and recursive rewriting to ensure correctness and completeness of the transducer for all valid inputs.

Experimental results

Research questions

• RQ1 Does the negative base $-\beta$, where $\beta$ is the real root of $x^3 - mx^2 - mx - m$, satisfy the finiteness property (-F)?
• RQ2 Can an explicit finite state transducer be constructed to perform addition and subtraction in the negative Tribonacci base $-\gamma$?
• RQ3 What is the maximal number of fractional digits that can arise from adding or subtracting two $(-\gamma)$-integers?
• RQ4 Is there a generalization of the finiteness property to higher-degree negative Pisot bases with symmetric coefficients?

Key findings

• The finiteness property (-F) holds for all negative bases $-\beta$ where $\beta$ is the real root of $x^3 - mx^2 - mx - m$ for $m \in \mathbb{N}$, meaning the set of finite $(-\beta)$-expansions forms a ring.
• An explicit finite state transducer with 13 states is constructed to perform addition and subtraction in the negative Tribonacci base $-\gamma$, where $\gamma$ is the real root of $x^3 - x^2 - x - 1$.
• The transducer correctly handles all valid inputs and terminates for any input ending in $0^\omega$, ensuring finite computation.
• The maximal number of fractional digits arising from addition or subtraction of two $(-\gamma)$-integers is bounded by 5.
• The transducer’s structure confirms that the number of fractional digits is uniformly bounded, which supports the ring property and enables algorithmic verification.
• The construction provides a concrete, constructive alternative to the non-constructive proof of transducer existence in prior work, enabling practical implementation and analysis.