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arXiv · 2503.16330

If a machine did it, it is probably transcendental (even $p$-adically)

Laura Capuano·Sara Checcoli·Marzio Mula·Lea Terracini
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Abstract

Continued fraction expansions provide a well-established bridge between algebraic properties of numbers and combinatorics on words. In this article, we investigate the algebraicity of $p$-adic numbers whose continued fractions arise from certain classes of words which generalize the classical automatic, periodic and palindromic words. Our main result shows that, under mild conditions on the $p$-adic continued fraction expansion, such numbers are either algebraic of degree at most 2 or transcendental. This result provides an analogue of results of Bugeaud and Adamczewski-Bugeaud in the real setting and extends previous works that were limited to specific choices of $p$-adic floor functions and less general classes of words.

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Laura Capuano, Sara Checcoli, Marzio Mula, Lea Terracini. 2025-03-20. If a machine did it, it is probably transcendental (even $p$-adically). https://arxiv.org/abs/2503.16330

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