Hausdorff dimension of real numbers with bounded digit averages

Cesaratto, Eda; Vallée, Brigitte

doi:10.4064/aa125-2-2

Cited by 13 publications

(7 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A crucial point is the US Property for the quasi-inverse (Id − H s ) −1 of the (plain) transfer operator [there exists a vertical strip where the quasi-inverse has a unique pole and is of polynomial growth for s → ∞]. Thus, we need to extend this type of the results to the quasi-inverse (Id − H M,s ) −1 of the restricted operator, with estimates uniform with respect to M. We mainly use perturbation theory (since the operator H M,s is a small perturbation of H s , when M → ∞), and estimate the speed of convergence of the spectral objects of H M,s to those of H s by extending to our present framework methods due to Cesaratto and Vallée [3] and Hensley [12].…”

Section: Motivations and Methodsmentioning

confidence: 99%

Small quotients in Euclidean algorithms

Cesaratto

Vallée

2010

Ramanujan J

Self Cite

View full text Add to dashboard Cite

Numbers whose continued fraction expansion contains only small digits have been extensively studied. In the real case, the Hausdorff dimension σ M of the reals with digits in their continued fraction expansion bounded by M was considered, and estimates of σ M for M → ∞ were provided by Hensley (the case of a fixed bound M when the denominator N tends to ∞. Later, Hensley (Pac. J. Math. 151(2):237-255, 1991) dealt with the case of a bound M which may depend on the denominator N, and obtained a precise estimate on the cardinality of rational numbers of denominator less than N whose digits (in the continued fraction expansion) are less than M(N), provided the bound M(N) is large enough with respect to N. This paper improves this last result of Hensley towards four directions. First, it considers various continued fraction expansions; second, it deals with various probability settings (and not only the uniform probability); third, it studies the case of all possible sequences M(N), with the only restriction that M(N) is at least equal to a given constant M

show abstract

Section: Motivations and Methodsmentioning

confidence: 99%

Small quotients in Euclidean algorithms

Cesaratto

Vallée

2010

Ramanujan J

Self Cite

View full text Add to dashboard Cite

show abstract

“…In [3], Cesaratto and Vallée gave an exhaustive investigation of the sets of continued fractions whose arithmetic means are all bounded by M > 0. Recently, Iommi and Jordan [10] extensively studied the Hausdorff dimension of the level sets of reals whose arithmetic means of partial quotients have finite limits by the theory of dynamical system.…”

Section: Introductionmentioning

confidence: 99%

On the distribution for sums of partial quotients in continued fraction expansions

Wu¹,

Xu²

2011

Nonlinearity

View full text Add to dashboard Cite

Let x ∈ [0, 1) and [a 1 (x), a 2 (x), . . .] be the continued fraction expansion of x. For any n 1, write S n (x) = n k=1 a k (x). Khintchine (1935 Compos. Math. 1 361-82) proved that S n (x) n log n converges in measure to 1 log 2 with respect to L 1 , where L 1 denotes the one dimensional Lebesgue measure. Philipp (1988 Monatsh. Math. 105 195-206) showed that there is not a reasonable normalizing sequence such that a strong law of large numbers is satisfied. In this paper, we show that for any α 0, the setis of Hausdorff dimension 1. Furthermore, we prove that the Hausdorff dimension of the set consisting of reals whose sums of partial quotients grow at a given polynomial rate is 1.

show abstract

“…During his thesis, Hervé Daudé made experiments providing evidence for the Gaussian limit property of the number of steps. Joint work with Eda Cesaratto [10] involved an extensive use of the weighted transfer operator, and some of the ideas that we shared on that occasion proved very useful for the present paper. We have had many stimulating discussions with Philippe Flajolet regarding the saddle-point method and the notion of smoothed costs.…”

mentioning

confidence: 99%

Euclidean algorithms are Gaussian

Baladi

Vallée

2005

Journal of Number Theory

Self Cite

307

View full text Add to dashboard Cite

We obtain a central limit theorem for a general class of additive parameters (costs, observables) associated to three standard Euclidean algorithms, with optimal speed of convergence. We also provide very precise asymptotic estimates and error terms for the mean and variance of such parameters. For costs that are lattice (including the number of steps), we go further and establish a local limit theorem, with optimal speed of convergence. We view an algorithm as a dynamical system restricted to rational inputs, and combine tools imported from dynamics, such as transfer operators, with various other techniques: Dirichlet series, Perron's formula, quasi-powers theorems, and the saddle-point method. Such dynamical analyses had previously been used to perform the average-case analysis of algorithms. For the present (dynamical) analysis in distribution, we require estimates on transfer operators when a parameter varies along vertical lines in the complex plane. To prove them, we adapt techniques introduced recently by Dolgopyat in the context of continuous-time dynamics (Ann. Math. 147 (1998) 357).

show abstract

Hausdorff dimension of real numbers with bounded digit averages

Cited by 13 publications

References 39 publications

Small quotients in Euclidean algorithms

Small quotients in Euclidean algorithms

On the distribution for sums of partial quotients in continued fraction expansions

Euclidean algorithms are Gaussian

Contact Info

Product

Resources

About