Results 1 
7 of
7
The Dimensions of Individual Strings and Sequences
 INFORMATION AND COMPUTATION
, 2003
"... A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary ..."
Abstract

Cited by 93 (10 self)
 Add to MetaCart
A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary) sequence S a dimension, which is a real number dim(S) in the interval [0, 1]. Sequences that
On Khintchine exponents and Lyapunov exponents of continued fractions, Ergod
 Th. Dynam. Sys
"... Abstract. Assume that x ∈ [0,1) admits its continued fraction expansion x = [a1(x), a2(x), · · ·]. The Khintchine exponent γ(x) of x is defined by 1 Pn γ(x): = lim n→ ∞ n j=1 log aj(x) when the limit exists. Khintchine spectrum dim Eξ is fully studied, where Eξ: = {x ∈ [0, 1) : γ(x) = ξ} (ξ ≥ 0) ..."
Abstract

Cited by 6 (5 self)
 Add to MetaCart
Abstract. Assume that x ∈ [0,1) admits its continued fraction expansion x = [a1(x), a2(x), · · ·]. The Khintchine exponent γ(x) of x is defined by 1 Pn γ(x): = lim n→ ∞ n j=1 log aj(x) when the limit exists. Khintchine spectrum dim Eξ is fully studied, where Eξ: = {x ∈ [0, 1) : γ(x) = ξ} (ξ ≥ 0) and dim denotes the Hausdorff dimension. In particular, we prove the remarkable fact that the Khintchine spectrum dim Eξ, as function of ξ ∈ [0,+∞), is neither concave nor convex. This is a new phenomenon from the usual point of view of multifractal analysis. Fast Khintchine exponents defined by γϕ 1 Pn (x): = lim n→ ∞ ϕ(n) j=1 log aj(x) are also studied, where ϕ(n) tends to the infinity faster than n does. Under some regular conditions on ϕ, it is proved that the fast Khintchine spectrum dim({x ∈ [0,1] : γϕ (x) = ξ}) is a constant function. Our method also works for other spectra like the Lyapunov spectrum and the fast Lyapunov spectrum.
The arithmeticgeometric scaling spectrum for continued fractions
 Arkivför Matematik
"... Abstract. To compare the continued fraction digits with the denominators of the corresponding approximants we introduce the arithmeticgeometric scaling. To determine its multifractal spectrum completely we impose a number theoretical free energy function and show that the Hausdor dimension of sets ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Abstract. To compare the continued fraction digits with the denominators of the corresponding approximants we introduce the arithmeticgeometric scaling. To determine its multifractal spectrum completely we impose a number theoretical free energy function and show that the Hausdor dimension of sets consisting of irrationals with the same scaling exponent coincides with the Legendre transform of this free energy function. Furthermore, we determine the asymptotic of the local behaviour of the spectrum at the right boundary point and discuss a connection to the set of irrationals with continued fraction digits exceeding a given number which tends to in nity.
1 On the Hausdorff dimension of certain sets arising in Number Theory
, 1999
"... Any real number x in the unit interval can be expressed as a continued fraction x = [n1,...,n,...]. Subsets of zero measure are obtained by imposing simple condiN tions on the n. By imposing n ≤ m ∀ N ∈ IN, Jarnik defined the corresponding N N sets Em and gave a first estimate of dH(Em), dH the Hau ..."
Abstract
 Add to MetaCart
Any real number x in the unit interval can be expressed as a continued fraction x = [n1,...,n,...]. Subsets of zero measure are obtained by imposing simple condiN tions on the n. By imposing n ≤ m ∀ N ∈ IN, Jarnik defined the corresponding N N sets Em and gave a first estimate of dH(Em), dH the Hausdorff dimension. Subsequent authors improved these estimates. In this paper we deal with dH(Em) and ∑N i=1 dH(Fm), Fm being the set of real numbers for which ni N ≤ m.
DIOPHANTINE PROPERTIES OF MEASURES INVARIANT WITH RESPECT TO THE GAUSS MAP
"... Diophantine properties of probability measures invariant under the Gauss map. Specifically, we prove that every such measure which has finite Lyapunov exponent is extremal, i.e. gives zero measure to the set of very well approximable numbers. We show on the other hand that there exist examples where ..."
Abstract
 Add to MetaCart
Diophantine properties of probability measures invariant under the Gauss map. Specifically, we prove that every such measure which has finite Lyapunov exponent is extremal, i.e. gives zero measure to the set of very well approximable numbers. We show on the other hand that there exist examples where the Lyapunov exponent is infinite and the invariant measure is not extremal. We provide a partial converse to B. Weiss’s result of Khinchine type [12] by constructing a large class of measures, which are both conformal and Ahlfors regular, and for which the divergence of Weiss’s series entails the ψapproximability of almost all numbers. 1.
associated with the
, 2008
"... A formula for the fractal dimension d ∼ 0.87 of the Cantorian set underlying the Devil’s staircase ..."
Abstract
 Add to MetaCart
A formula for the fractal dimension d ∼ 0.87 of the Cantorian set underlying the Devil’s staircase
THÈSE En vue de l’obtention du Grade de Docteur de l’Université de Picardie
"... Mes tous premiers remerciements vont à mon Directeur de thèse AiHua Fan, qui a su me faire découvrir de nombreux domaines passionnants des mathématiques. Sans ses encouragements constants et ses judicieux conseils, cette thèse n’aurait pu aboutir. Ses qualités humaines et scientifiques demeurent gr ..."
Abstract
 Add to MetaCart
Mes tous premiers remerciements vont à mon Directeur de thèse AiHua Fan, qui a su me faire découvrir de nombreux domaines passionnants des mathématiques. Sans ses encouragements constants et ses judicieux conseils, cette thèse n’aurait pu aboutir. Ses qualités humaines et scientifiques demeurent gravées dans ma mémoire. Il m’a beaucoup appris. Je ne pourrais jamais lui témoigner suffisamment ma gratitude.