Transcendence for Pisot Morphic Words over an Algebraic Base

2024-05-06

AI Summary

This paper investigates the transcendence of real numbers represented by β-expansions, where β is an algebraic Pisot number and the digit sequences are generated by irreducible Pisot morphisms. For any such sequence (u), the associated real number ([u]_eta) is proven to lie either in (mathbb{Q}(eta)) or to be transcendental—establishing a sharp rational–transcendental dichotomy. Methodologically, the work extends the rational–transcendental dichotomy previously known for uniform morphic sequences to the irreducible Pisot case; under the Pisot conjecture, this extension holds over arbitrary finite alphabets. By integrating tools from algebraic number theory, symbolic dynamics, and spectral analysis, the paper provides unconditional transcendence proofs for ([u]_eta) when (u) is a (k)-bonacci word ((k geq 2)). The key innovation lies in removing the uniformity constraint, yielding a broader transcendence criterion for Pisot morphic sequences—thereby opening new avenues for studying the arithmetic properties of automatic sequences and advancing the Pisot conjecture.

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• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

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Abstract

It is known that for a uniform morphic sequence $oldsymbol u = langle u_n angle_{n=0}^infty$ and an algebraic number $eta$ such that $|eta|>1$, the number $[![oldsymbol{u} ]!]_eta:=sum_{n=0}^infty frac{u_n}{eta^n}$ either lies in $mathbb Q(eta)$ or is transcendental. In this paper we show a similar rational-transcendental dichotomy for sequences defined by irreducible Pisot morphisms. Subject to the Pisot conjecture (an irreducible Pisot morphism has pure discrete spectrum), we generalise the latter result to arbitrary finite alphabets. In certain cases we are able to show transcendence of $[![oldsymbol{u}]!]_{eta}$ outright. In particular, for $kgeq 2$, if $oldsymbol u$ is the $k$-bonacci word then $[![oldsymbol{u}]!]_{eta}$ is transcendental.

Problem

• Dichotomy for Pisot morphic sequences in algebraic bases
• Generalization to arbitrary alphabets under Pisot conjecture
• Transcendence proof for k-bonacci word expansions

Innovation

• Uses irreducible Pisot morphisms for sequences
• Generalizes results to arbitrary finite alphabets
• Proves transcendence for k-bonacci words