Irrationality measures for some automatic real numbers

نویسندگان

  • Boris Adamczewski
  • Tanguy Rivoal
چکیده

This paper is devoted to the rational approximation of automatic real numbers, that is, real numbers whose expansion in an integer base can be generated by a finite automaton. We derive upper bounds for the irrationality exponent of famous automatic real numbers associated with the Thue–Morse, Rudin–Shapiro, paperfolding and Baum–Sweet sequences. These upper bounds arise from the construction of some explicit Padé or Padé type approximants for the generating functions of these sequences. In particular, we prove that the Thue–Morse–Mahler numbers have an irrationality exponent at most equal to 4. We also obtain an explicit description of infinitely many convergents to these numbers.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Diophantine Properties of Automatic Real Numbers

We study some diophantine properties of automatic real numbers and we present a method to derive irrationality measures for such numbers. As a consequence, we prove that the b-adic expansion of a Liouville number cannot be generated by a finite automaton, a conjecture due to Shallit.

متن کامل

Diophantine properties of real numbers generated by finite automata

We study some Diophantine properties of automatic real numbers and we present a method to derive irrationality measures for such numbers. As a consequence, we prove that the b-adic expansion of a Liouville number cannot be generated by a finite automaton, a conjecture due to Shallit.

متن کامل

A devil’s staircase from rotations and irrationality measures for Liouville numbers∗

From Sturmian and Christoffel words we derive a strictly increasing function ∆ : [0,∞) → R. This function is continuous at every irrational point, while at rational points, left-continuous but not right-continuous. Moreover, it assumes algebraic integers at rationals, and transcendental numbers at irrationals. We also see that the differentiation of ∆ distinguishes some irrationality measures o...

متن کامل

Irrationality Measures of log 2 and π/√3

1. Irrationality Measures An irrationality measure of x ∈ R \Q is a number μ such that ∀ > 0,∃C > 0,∀(p, q) ∈ Z, ∣∣∣∣x− pq ∣∣∣∣ ≥ C qμ+ . This is a way to measure how well the number x can be approximated by rational numbers. The measure is effective when C( ) is known. We denote inf {μ | μ is an irrationality measure of x } by μ(x), and we call it the irrationality measure of x. By definition,...

متن کامل

An essay on irrationality measures of π and other logarithms

produces ‘good’ rational approximations to log a. There are several ways of performing integration in (1) in order to show that the integral lies in Q log a+ Q; we give an exposition of different methods below. The aim of this essay is to demonstrate how suitable generalizations of the integrals in (1) allow to prove the best known results on irrationality measures of the numbers log 2, π and l...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2008