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On the expansion of some exponential periods in an integer base
 MATHEMATISCHE ANNALEN
, 2010
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A devil’s staircase from rotations and irrationality measures for Liouville numbers
, 2007
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Continued fractions with low complexity: Transcendence measures and quadratic approximation
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THE FIRST RETURN TIME PROPERTIES OF AN IRRATIONAL ROTATION
, 2008
"... If an ergodic system has positive entropy, then the ShannonMcMillanBreiman theorem provides a relationship between the entropy and the size of an atom of the iterated partition. The system also has OrnsteinWeiss’ first return time property, which offers a method of computing the entropy via an or ..."
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If an ergodic system has positive entropy, then the ShannonMcMillanBreiman theorem provides a relationship between the entropy and the size of an atom of the iterated partition. The system also has OrnsteinWeiss’ first return time property, which offers a method of computing the entropy via an orbit. We consider irrational rotations which are the simplest model of zero entropy. We prove that almost every irrational rotation has the analogous properties if properly normalized. However there are some irrational rotations that exhibit different behavior.
An Irrationality Measure for Regular Paperfolding Numbers
"... Let F(z) = ∑ n�1 fnzn be the generating series of the regular paperfolding sequence. For a real number α the irrationality exponent µ(α), of α, is defined as the supremum of the set of real numbers µ such that the inequality α − p/q  < q−µ has infinitely many solutions (p, q) ∈ Z × N. In thi ..."
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Let F(z) = ∑ n�1 fnzn be the generating series of the regular paperfolding sequence. For a real number α the irrationality exponent µ(α), of α, is defined as the supremum of the set of real numbers µ such that the inequality α − p/q  < q−µ has infinitely many solutions (p, q) ∈ Z × N. In this paper, using a method introduced by Bugeaud, we prove that µ(F(1/b)) � 275331112987 = 2.002075359 · · · 137522851840 for all integers b � 2. This improves upon the previous bound of µ(F(1/b)) � 5 given by Adamczewski and Rivoal.
WORDS AND TRANSCENDENCE
, 2009
"... Abstract. Is it possible to distinguish algebraic from transcendental real numbers by considering the bary expansion in some base b � 2? In 1950, É. Borel suggested that the answer is no and that for any real irrational algebraic number x and for any base g � 2, the gary expansion of x should sati ..."
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Abstract. Is it possible to distinguish algebraic from transcendental real numbers by considering the bary expansion in some base b � 2? In 1950, É. Borel suggested that the answer is no and that for any real irrational algebraic number x and for any base g � 2, the gary expansion of x should satisfy some of the laws that are shared by almost all numbers. For instance, the frequency where a given finite sequence of digits occurs should depend only on the base and on the length of the sequence. We are very far from such a goal: there is no explicitly known example of a triple (g, a, x), where g � 3 is an integer, a a digit in {0,...,g − 1} and x a real irrational algebraic number, for which one can claim that the digit a occurs infinitely often in the gary expansion of x. Hence there is a huge gap between the established theory and the expected state of the art. However, some progress has been made recently, thanks mainly to clever use of Schmidt’s subspace theorem. We review some of these results. 1. Normal Numbers and Expansion of Fundamental Constants 1.1. Borel and Normal Numbers. In two papers, the first [28] published in 1909 and the second [29] in 1950, Borel studied the gary expansion of real numbers, where g � 2 is a positive integer. In his second paper, he suggested that this expansion for a real irrational algebraic number should satisfy some of the laws shared by almost all numbers, in the sense of Lebesgue measure. Let g � 2 be an integer. Any real number x has a unique expansion x = a−kg k + ···+ a−1g + a0 + a1g −1 + a2g −2 + ·· ·, where k � 0 is an integer and the ai for i � −k, namely the digits of x in the expansion in base g of x, belong to the set {0, 1,...,g − 1}. Uniqueness is subject to the condition that the sequence (ai)i�−k is not ultimately constant and equal to g − 1. We write this expansion x = a−k ···a−1a0.a1a2 ·· ·. in base 10 (decimal expansion), whereas
Morphic and Automatic Words: Maximal Blocks and Diophantine Approximation
, 2011
"... Let w be a morphic word over a finite alphabet Σ, and let ∆ be a nonempty subset of Σ. We study the behavior of maximal blocks consisting only of letters from ∆ in w, and prove the following: let (ik, jk) denote the starting and ending positions, respectively, of the k’th maximal ∆block in w. Then ..."
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Let w be a morphic word over a finite alphabet Σ, and let ∆ be a nonempty subset of Σ. We study the behavior of maximal blocks consisting only of letters from ∆ in w, and prove the following: let (ik, jk) denote the starting and ending positions, respectively, of the k’th maximal ∆block in w. Then lim supk→∞(jk/ik) is algebraic if w is morphic, and rational if w is automatic. As a result, we show that the same conclusion holds if (ik, jk) are the starting and ending positions of the k’th maximal zero 1 block, and, more generally, of the k’th maximal xblock, where x is an arbitrary word. This enables us to draw conclusions about the irrationality exponent of automatic and morphic numbers. In particular, we show that the irrationality exponent of automatic (resp., morphic) numbers belonging to a certain class that we define is rational (resp., algebraic). 1