Hausdorff dimension of sets arising in number theory

Bumby, Richard T.

doi:10.1007/bfb0074599

Cited by 20 publications

(24 citation statements)

References 9 publications

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“…It is of considerable interest (see [2,3]) to allow maps θ i which may not be affine linear. For example, in studying subsets of R defined by properties of their continued fraction expansions, one is led to maps θ i : [ and if θ i (C) and θ j (C) are disjoint for 1 ≤ i < j ≤ N , we shall obtain below a formula for the Hausdorff dimension of C. In fact, we shall obtain such a formula in a setting similar to that of Mauldin and Williams, but using contractions and infinitesimal similitudes, rather than affine linear contractions which are similitudes.…”

Section: Introductionmentioning

confidence: 99%

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Positive operators and Hausdorff dimension of invariant sets

Nussbaum

Priyadarshi

Lunel

2012

Trans. Amer. Math. Soc.

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Abstract. In this paper we obtain theorems which give the Hausdorff dimension of the invariant set for a finite family of contraction mappings which are "infinitesimal similitudes" on a complete, perfect metric space. Our work generalizes the graph-directed construction of Mauldin and Williams (1988) and is related in its general setting to results of Schief (1996), but differs crucially in that the mappings need not be similitudes. We use the theory of positive linear operators and generalizations of the Krein-Rutman theorem to characterize the Hausdorff dimension as the unique value of σ > 0 for which r(L σ ) = 1, where L σ , σ > 0, is a naturally associated family of positive linear operators and r(L σ ) denotes the spectral radius of L σ . We also indicate how these results can be generalized to countable families of infinitesimal similitudes. The intent here is foundational: to derive a basic formula in its proper generality and to emphasize the utility of the theory of positive linear operators in this setting. Later work will explore the usefulness of the basic theorem and its functional analytic setting in studying questions about Hausdorff dimension.

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Section: Introductionmentioning

confidence: 99%

“…2 , which implies m + x = 1, i.e., x = 1 − m. Since m ≥ 1 and x ≥ 0 for z ∈ X, it follows that m = 1 and x = 0. But x = 0 implies that z = 0.…”

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confidence: 98%

Positive operators and Hausdorff dimension of invariant sets

Nussbaum

Priyadarshi

Lunel

2012

Trans. Amer. Math. Soc.

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“…The particular case of "constrained" continued fractions was extensively studied; the beginners were Jarník [28,29], Besicovitch [5] and Good [17]. Then Cusick [11], Hirst [24] and Bumby [9] brought important contributions, and finally Hensley [19][20][21][22] completely solved the problem. In [41], this result was extended to the case of "periodic" constraints.…”

Section: )mentioning

confidence: 99%

Hausdorff dimension of real numbers with bounded digit averages

Cesaratto

Vallée

2006

Acta Arith.

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“…Unfortunately, the sum defining`(2 n , s, A) has |A| 2 n terms. Until recently, this doubly exponential explosion in complexity as a function of the number of digits obtained was inherent in all known methods for estimating dim E A [3,8,13]. Not surprisingly, the best that could be achieved was the modest dim E [1,2] # (0.5312, 0.5314).…”

Section: Introductionmentioning

confidence: 96%

A Polynomial Time Algorithm for the Hausdorff Dimension of Continued Fraction Cantor Sets

Hensley

1996

Journal of Number Theory

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For any finite set A of positive integers, let E A :=[: # (0, 1): : is irrational, and every partial quotient in the (infinite) simple continued fraction expansion of : is an element of A]. For sets A with fewer than two elements, E A is uninteresting. For |A| 2, E A is a kind of Cantor fractal dust, with a Hausdorff dimension (dim E A ) between 0 and 1. This work presents an algorithm which, given a finite set A of between 2 and N positive integers 2 N , determines dim E A to within \2 &N using O(N 7 ) elementary bit operations. There is also a convenient implementation of the algorithm in Mathematica code, together with a small table and some conjectures.

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Hausdorff dimension of sets arising in number theory

Cited by 20 publications

References 9 publications

Positive operators and Hausdorff dimension of invariant sets

Positive operators and Hausdorff dimension of invariant sets

Hausdorff dimension of real numbers with bounded digit averages

A Polynomial Time Algorithm for the Hausdorff Dimension of Continued Fraction Cantor Sets

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