Transcendence for Pisot Morphic Words over an Algebraic Base

TL;DR

This paper establishes a dichotomy for sums of Pisot morphic sequences over algebraic bases, showing they are either algebraic or transcendental, and proves transcendence for specific cases like the k-bonacci word.

Contribution

It extends the rational-transcendental dichotomy to irreducible Pisot morphic sequences and proves transcendence for certain well-known sequences.

Findings

Sum $[oldsymbol{u}]_eta$ is either algebraic or transcendental.

Transcendence is proven for the k-bonacci word.

Results depend on the Pisot conjecture, assuming pure discrete spectrum.

Abstract

It is known that for a uniform morphic sequence $\boldsymbol u = \langle u_n\rangle_{n=0}^\infty$ and an algebraic number $\beta$ such that $|\beta|>1$, the number $[\![\boldsymbol{u} ]\!]_\beta:=\sum_{n=0}^\infty \frac{u_n}{\beta^n}$ either lies in $\mathbb Q(\beta)$ 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 $[\![\boldsymbol{u}]\!]_{\beta}$ outright. In particular, for $k\geq 2$, if $\boldsymbol u$ is the $k$-bonacci word then $[\![\boldsymbol{u}]\!]_{\beta}$ is transcendental.