number theory

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

arXiv:2503.16330

summary

The paper proves that p‑adic numbers whose continued‑fraction expansions come from broad classes of combinatorial words are either quadratic algebraic or transcendental, extending known real‑number results to the p‑adic setting.

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 -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 -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 -adic floor functions and less general classes of words.

33 pages, accepted for publication on Math. Z

Topics & keywords

#p-adic numbers#continued fractions#transcendence#automatic sequences#combinatorics on words#algebraic degreep-adic continued fractionstranscendental numbersautomatic wordsquadratic algebraicBugeaudAdamczewski
If a machine did it, it is probably transcendental (even $p$-adically) · wovepaper