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2019•Journal of the Australian Mathematical SocietyRequires access

ON SPRINDŽUK’S CLASSIFICATION OF -ADIC NUMBERS

Yann Bugeaud, Gülcan Kekeç

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Abstract

Abstract We carry Sprindžuk’s classification of the complex numbers to the field $\mathbb{Q}_{p}$ of $p$ -adic numbers. We establish several estimates for the $p$ -adic distance between $p$ -adic roots of integer polynomials, which we apply to show that almost all $p$ -adic numbers, with respect to the Haar measure, are $p$ -adic $\tilde{S}$ -numbers of order 1.

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What this paper is about

Abstract We carry Sprindžuk’s classification of the complex numbers to the field $\mathbb{Q}_{p}$ of $p$ -adic numbers. We establish several estimates for the $p$ -adic distance between $p$ -adic roots of integer polynomials, which we apply to show that almost all $p$ -adic numbers, with respect to the Haar measure, are $p$ -adic $\tilde{S}$ -numbers of order 1.

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Available abstract

Abstract We carry Sprindžuk’s classification of the complex numbers to the field $\mathbb{Q}_{p}$ of $p$ -adic numbers. We establish several estimates for the $p$ -adic distance between $p$ -adic roots of integer polynomials, which we apply to show that almost all $p$ -adic numbers, with respect to the Haar measure, are $p$ -adic $\tilde{S}$ -numbers of order 1.

Key concepts: Mathematics, Integer (computer science), Order (exchange), Haar, Carry (investment), Combinatorics, Real number, Algebraic number field

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