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24
On the complexity of algebraic numbers I. Expansions in integer bases
, 2005
"... Let b ≥ 2 be an integer. We prove that the badic expansion of every irrational algebraic number cannot have low complexity. Furthermore, we establish that irrational morphic numbers are transcendental, for a wide class of morphisms. In particular, irrational automatic numbers are transcendental. O ..."
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Cited by 47 (23 self)
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Let b ≥ 2 be an integer. We prove that the badic expansion of every irrational algebraic number cannot have low complexity. Furthermore, we establish that irrational morphic numbers are transcendental, for a wide class of morphisms. In particular, irrational automatic numbers are transcendental. Our main tool is a new, combinatorial transcendence criterion.
On two notions of complexity of algebraic numbers
, 2007
"... We derive new, improved lower bounds for the block complexity of an irrational algebraic number and for the number of digit changes in the bary expansion of an irrational algebraic number. To this end, we apply a version of the Quantitative Subspace Theorem by Evertse and Schlickewei [14], Theore ..."
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Cited by 8 (8 self)
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We derive new, improved lower bounds for the block complexity of an irrational algebraic number and for the number of digit changes in the bary expansion of an irrational algebraic number. To this end, we apply a version of the Quantitative Subspace Theorem by Evertse and Schlickewei [14], Theorem 2.1.
On the complexity of algebraic numbers II. Continued fractions
 Acta Math
"... Let b ≥ 2 be an integer. Émile Borel [9] conjectured that every real irrational algebraic number α should satisfy some of the laws shared by almost all real numbers with respect to their badic expansions. Despite some recent progress [1, 3, 7], we are still very far away from establishing such a st ..."
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Cited by 8 (2 self)
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Let b ≥ 2 be an integer. Émile Borel [9] conjectured that every real irrational algebraic number α should satisfy some of the laws shared by almost all real numbers with respect to their badic expansions. Despite some recent progress [1, 3, 7], we are still very far away from establishing such a strong result. In the present work, we are concerned
Normal Numbers and Pseudorandom Generators
, 2011
"... For an integer b ≥ 2 a real number α is bnormal if, for all m> 0, every mlong string of digits in the baseb expansion of α appears, in the limit, with frequency b −m. Although almost all reals in [0, 1] are bnormal for every b, it has been rather difficult to exhibit explicit examples. No res ..."
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Cited by 6 (2 self)
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For an integer b ≥ 2 a real number α is bnormal if, for all m> 0, every mlong string of digits in the baseb expansion of α appears, in the limit, with frequency b −m. Although almost all reals in [0, 1] are bnormal for every b, it has been rather difficult to exhibit explicit examples. No results whatsoever are known, one way or the other, for the class of “natural ” mathematical constants, such as π, e, √ 2 and log 2. In this paper, we summarize some previous normality results for a certain class of explicit reals, and then show that a specific member of this class, while provably 2normal, is provably not 6normal. We then show that a practical and reasonably effective pseudorandom number generator can be defined based on the binary digits of this constant, and conclude by sketching out some directions for further research.
On the bits counting function of real numbers
 J. AUSTRAL. MATH. SOC
"... Let Bn(x) denote the number of 1’s occuring in the binary expansion of an irrational number x> 0. A difficult problem is to provide nontrivial lower bounds for Bn(x) for interesting number numbers like 2, e or pi: their conjectural simple normality in base 2 is equivalent to Bn(x) ∼ n/2. In th ..."
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Cited by 4 (0 self)
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Let Bn(x) denote the number of 1’s occuring in the binary expansion of an irrational number x> 0. A difficult problem is to provide nontrivial lower bounds for Bn(x) for interesting number numbers like 2, e or pi: their conjectural simple normality in base 2 is equivalent to Bn(x) ∼ n/2. In this article, amongst other things, we prove inequalities relating Bn(x+y), Bn(xy) and Bn(1/x) to Bn(x) and Bn(y) for any irrational numbers x, y> 0, which we prove to be sharp up to a multiplicative constant. As a byproduct, we provide an answer to a question raised by Bailey et al. (On the binary expansions of algebraic numbers, J. Théor. Nombres Bordeaux 16 (2004), no. 3, 487–518) concerning the binary digits of the square of a series related to the Fibonacci sequence. We also obtain a slight refinement of the main theorem of the same article, which provides nontrivial lower bound for Bn(α) for any real irrational algebraic number. We conclude the article with effective or conjectural lower bounds for Bn(x) when x is a transcendental number.
Fifty years of the spectrum problem: survey and new results
, 2009
"... In 1952, Heinrich Scholz published a question in the Journal of Symbolic Logic asking for a characterization of spectra, i.e., sets of natural numbers that are the cardinalities of finite models of first order sentences. Günter Asser asked whether the complement of a spectrum is always a spectrum. ..."
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In 1952, Heinrich Scholz published a question in the Journal of Symbolic Logic asking for a characterization of spectra, i.e., sets of natural numbers that are the cardinalities of finite models of first order sentences. Günter Asser asked whether the complement of a spectrum is always a spectrum. These innocent questions turned out to be seminal for the development of finite model theory and descriptive complexity. In this paper we survey developments over the last 50odd years pertaining to the spectrum problem. Our presentation follows conceptual developments rather than the chronological order. Originally a number theoretic problem, it has been approached in terms of recursion theory, resource bounded complexity theory, classification by complexity of the defining sentences, and finally in terms of structural graph theory. Although Scholz’ question was answered in various ways, Asser’s question remains open. One appendix paraphrases the contents of several early and not easily accessible papers by G. Asser, A. Mostowski, J. Bennett and S. Mo. Another appendix contains a compendium of questions and conjectures which remain open.
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
Convergents and irrationality measures of logarithms
 Rev. Mat. Iberoamericana
"... Abstract. We prove new irrationality measures with restricted denominators of the form dsbνmcB m (where B,m ∈ N, ν> 0, s ∈ {0, 1} and dm = lcm{1, 2,...,m}) for values of the logarithm at certain rational numbers r> 0. In particular, we show that such an irrationality measure of log(r) is arbit ..."
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Abstract. We prove new irrationality measures with restricted denominators of the form dsbνmcB m (where B,m ∈ N, ν> 0, s ∈ {0, 1} and dm = lcm{1, 2,...,m}) for values of the logarithm at certain rational numbers r> 0. In particular, we show that such an irrationality measure of log(r) is arbitrarily close to 1 provided r is sufficiently close to 1. This implies certain results on the number of nonzero digits in the b–ary expansion of log(r) and on the structure of the denominators of convergents of log(r). No simple method for calculating the latter is known. For example, we show that, given integers a, c ≥ 1, for all large enough b, n, the denominator qn of the n–th convergent of log(1±a/b) cannot be written under the form dsbνmc(bc) m: this is true for a = c = 1, b ≥ 12 when s = 0, resp. b ≥ 2 when s = 1 and ν = 1. Our method rests on a detailed diophantine analysis of the upper Pade ́ table ([p/q])p≥q≥0 of the function log(1 − x). Finally, we remark that worse results (of these form) are currently provable for the exponential function, despite the fact that the complete Pade ́ table ([p/q])p,q≥0 of exp(x) and the convergents of exp(1/b), for b  ≥ 1, are wellknown, for example.
On the decimal expansion of algebraic numbers
, 2005
"... Fizikos ir matematikos fakulteto ..."
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