Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Transcendence of $p$-adic continued fractions and a quantitative $p$-adic Roth theorem

Anne Kalitzin, Nadir Murru

Abstract

In this paper, we improve some transcendence results for $p$--adic continued fractions. In particular, we prove that palindromic and quasi--periodic $p$--adic continued fractions converge either to transcendental numbers or quadratic irrationals, removing any restriction on the $p$--adic norm of the partial quotients (or convergents) considered in other works. Moreover, we provide a quantitative version of Ridout's theorem (the $p$--adic analogue of Roth's theorem), and we study the growth of denominators of convergents of algebraic numbers, establishing a $p$--adic version of a well--known result of Davenport and Roth.

Transcendence of $p$-adic continued fractions and a quantitative $p$-adic Roth theorem

Abstract

In this paper, we improve some transcendence results for --adic continued fractions. In particular, we prove that palindromic and quasi--periodic --adic continued fractions converge either to transcendental numbers or quadratic irrationals, removing any restriction on the --adic norm of the partial quotients (or convergents) considered in other works. Moreover, we provide a quantitative version of Ridout's theorem (the --adic analogue of Roth's theorem), and we study the growth of denominators of convergents of algebraic numbers, establishing a --adic version of a well--known result of Davenport and Roth.
Paper Structure (9 sections, 16 theorems, 157 equations)

This paper contains 9 sections, 16 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Diophantine approximation in Qp
  4. p--adic continued fractions
  5. Main results
  6. Palindromic p--adic continued fractions
  7. A quantitative version of p--adic Roth's theorem
  8. On the growth of denominators of convergents for algebraic numbers in Qp
  9. Quasi--periodic p--adic continued fractions

Key Result

Theorem A

Consider $\alpha \in \mathbb{Q}_{p}$ with $p$--adic continued fraction expansion $\alpha=\left[0, b_{1}, b_{2}, \ldots\right],$ where $\left(b_{i}\right)_{i \geq 1}$ is a sequence beginning with arbitrarily long palindromes. If $\max(|A_i|_\infty^{1/i}, |B_i|_\infty^{1/i}) < p^{1/4}$ for all $i\gg0$

Theorems & Definitions (26)

  • Theorem A
  • Theorem B
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 5
  • Lemma 6
  • proof
  • Theorem 7
  • ...and 16 more