Transcendence for Pisot Morphic Words over an Algebraic Base
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866918021500502016 |
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| author | Kebis, Pavol Luca, Florian Ouaknine, Joel Scoones, Andrew Worrell, James |
| author_facet | Kebis, Pavol Luca, Florian Ouaknine, Joel Scoones, Andrew Worrell, James |
| contents | It is known that for a uniform morphic sequence
$\boldsymbol u = \langle u_n\rangle_{n=0}^\infty$ and an algebraic
number $β$ such that $|β|>1$, the number
$[\![\boldsymbol{u} ]\!]_β:=\sum_{n=0}^\infty
\frac{u_n}{β^n}$ either lies in $\mathbb Q(β)$ 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
$[\![\boldsymbol{u}]\!]_β$ outright. In particular, for
$k\geq 2$, if $\boldsymbol u$ is the $k$-bonacci word then
$[\![\boldsymbol{u}]\!]_β$ is transcendental. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_05279 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Transcendence for Pisot Morphic Words over an Algebraic Base Kebis, Pavol Luca, Florian Ouaknine, Joel Scoones, Andrew Worrell, James Number Theory Formal Languages and Automata Theory 11J87, 11K16 F.0; F.4.3 It is known that for a uniform morphic sequence $\boldsymbol u = \langle u_n\rangle_{n=0}^\infty$ and an algebraic number $β$ such that $|β|>1$, the number $[\![\boldsymbol{u} ]\!]_β:=\sum_{n=0}^\infty \frac{u_n}{β^n}$ either lies in $\mathbb Q(β)$ 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 $[\![\boldsymbol{u}]\!]_β$ outright. In particular, for $k\geq 2$, if $\boldsymbol u$ is the $k$-bonacci word then $[\![\boldsymbol{u}]\!]_β$ is transcendental. |
| title | Transcendence for Pisot Morphic Words over an Algebraic Base |
| topic | Number Theory Formal Languages and Automata Theory 11J87, 11K16 F.0; F.4.3 |
| url | https://arxiv.org/abs/2405.05279 |