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Diophantine properties of real numbers generated by finite automata
 Compos. Math
"... Abstract. 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 badic expansion of a Liouville number cannot be generated by a finite automaton, a conjecture due to Shallit. 1. ..."
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Cited by 18 (4 self)
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Abstract. 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 badic expansion of a Liouville number cannot be generated by a finite automaton, a conjecture due to Shallit. 1.
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
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
bADIC NUMBERS IN PASCAL'S TRIANGLE MODULO b
"... h31 Form the product k 1 '(n kY by using this formula to expand k l and (n k) J '. Sum both sides and we get (83) S(iJ;n} ^t^^^jt^^tBAmAn k), r=0 ' d s=0 k = 0 which brings in a convolution of Bernoulli polynomials. Since the Bernoulli polynomials may be expressed in terms of Ber ..."
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of Bernoulli numbers by the further formula n (8.4) BAx) = X) {l) xn " " B ^ m = Q it would be possible to secure a convolution of the Bernoulli numbers. However, the author has not reduced this to any interesting or useful formula that appears to offer any advantages over those wev have derived
Factoring Generalized Repunits
"... Abstract. Twentyfive years ago, W. M. Snyder extended the notion of a repunit Rn to one in which for some positive integer b, Rn(b) has a badic expansion consisting of only ones. He then applied algebraic number theory in order to determine the pairs of integers under which Rn(b) has a prime divis ..."
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Abstract. Twentyfive years ago, W. M. Snyder extended the notion of a repunit Rn to one in which for some positive integer b, Rn(b) has a badic expansion consisting of only ones. He then applied algebraic number theory in order to determine the pairs of integers under which Rn(b) has a prime
An Alternative Construction of Normal Numbers
, 1998
"... A new class of b{adic normal numbers is built recursively by using Eulerian paths in a sequence of de{Bruijn digraphs. In this recursion, a path is constructed as an extension of the previous one, in such way that the b{adic block determined by the path contains the maximum number of dierent b{adic ..."
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Cited by 2 (1 self)
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xed integer. A number x 2 [0; 1] is a b{adic normal if each block q(1)q(2) q(k) on b symbols appears in the b{adic expansion of x with frequency b n . In [Bo] Borel proved that the set of all b{adic normals is a set of Lebesgue measure equals to 1, but it was only 35 years after Borel
Automatic Sequences and Related Topics Suites automatiques et sujets reliés
"... On real numbers generated by finite automata The badic expansion of any rational number is eventually periodic, but how regular or random (depending on the viewpoint) is the badic expansion of an irrational algebraic number? It seems that this natural question has been first addressed by Borel in ..."
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On real numbers generated by finite automata The badic expansion of any rational number is eventually periodic, but how regular or random (depending on the viewpoint) is the badic expansion of an irrational algebraic number? It seems that this natural question has been first addressed by Borel
Heterogenuous ubiquitous systems and Hausdorff dimension
"... Abstract. Let {xn}n∈N be a sequence of [0,1] d, {λn}n∈N a sequence of positive real numbers converging to 0, and δ> 1. The classical ubiquity results are concerned with the computation of the Hausdorff dimension of limsupsets of the form S(δ) = ⋂ ⋃ N∈N n≥N B(xn, λδn). Let µ be a positive Borel ..."
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Cited by 10 (8 self)
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of such sets opens the way to the study of several new objects and phenomena. Applications are given for the Diophantine approximation conditioned by (or combined with) badic expansion properties, by averages of some Birkhoff sums and branching random walks, as well as by asymptotic behavior of random
On the Hausdorff dimension of fractals given by certain expansions of real numbers
"... We transfer classical results on the Hausdorff dimension of badic and continued fraction expansions of real numbers to another expansion. ..."
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We transfer classical results on the Hausdorff dimension of badic and continued fraction expansions of real numbers to another expansion.
Results 1  10
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