Simon’s OPUC Hausdorff dimension conjecture
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DOI: 10.1007/s00208-021-02283-7
Abstract: We show that the Szegő matrices, associated with Verblunsky coefficients $\{\alpha_n\}_{n\in\mathbb{Z}_+}$ obeying $\sum_{n = 0}^\infty n^\gamma |\alpha_n|^2 < \infty$ for some $\gamma \in (0,1)$, are bounded for values $z \in \partial \mathbb{D}$ outside a set… read more here.
Keywords: dimension; hausdorff dimension; dimension conjecture; opuc hausdorff ... See more keywords
On the Hausdorff dimension of pinned distance sets
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DOI: 10.1007/s11856-019-1847-9
Abstract: We prove that if A is a Borel set in the plane of equal Hausdorff and packing dimension s > 1, then the set of pinned distances {|x − y| : y ∈ A} has… read more here.
Keywords: hausdorff dimension; dimension; dimension pinned; pinned distance ... See more keywords
Equilateral triangles in subsets of ℝd of large Hausdorff dimension
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DOI: 10.1007/s11856-019-1848-8
Abstract: We prove that subsets of ℝd, d ≥ 4 of large enough Hausdorff dimensions contain vertices of an equilateral triangle. It is known that additional hypotheses are needed to assure the existence of equilateral triangles… read more here.
Keywords: equilateral triangles; subsets large; triangles subsets; hausdorff dimension ... See more keywords
Hausdorff dimension regularity properties and games
Sign Up to like & getrecommendations! Published in 2022 at "Israel Journal of Mathematics"
DOI: 10.1007/s11856-022-2299-1
Abstract: The Hausdorff $\delta$-dimension game was introduced by Das, Fishman, Simmons and {Urba{\'n}ski} and shown to characterize sets in $\mathbb{R}^d$ having Hausdorff dimension $\leq \delta$. We introduce a variation of this game which also characterizes Hausdorff… read more here.
Keywords: hausdorff dimension; dimension leq; game; dimension ... See more keywords
Microlocal approach to the Hausdorff dimension of measures
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DOI: 10.1016/j.aim.2021.108088
Abstract: In this paper we study the dependence of geometric properties of Radon measures, such as Hausdorff dimension and rectifiability of singular sets, on the wavefront set. This is achieved by adapting the method of Brummelhuis… read more here.
Keywords: hausdorff dimension; microlocal approach; approach hausdorff; dimension ... See more keywords
The Hausdorff dimension of level sets described by Erdös–Rényi average
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DOI: 10.1016/j.jmaa.2017.09.018
Abstract: Abstract Let 0 ≤ α ≤ 1 and ϕ be an integer function defined on N ∖ { 0 } satisfying 1 ≤ ϕ ( n ) ≤ n . Define the level set E… read more here.
Keywords: dimension; hausdorff dimension; dimension level; sets described ... See more keywords
The Hausdorff Dimension is Convex on the Left Side of 1/4
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DOI: 10.1017/s0013091516000481
Abstract: Abstract Let d(c) denote the Hausdorff dimension of the Julia set Jc of the polynomial fc (z) = z 2 +c. The function c ↦ d(c) is real-analytic on the interval (–3/4, 1/4), which is… read more here.
Keywords: hausdorff dimension; dimension convex; convex left; left side ... See more keywords
Hausdorff dimension of frequency sets of univoque sequences
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DOI: 10.1080/14689367.2021.1905778
Abstract: We study the set consisting of univoque sequences, the set consisting of sequences in which the lengths of consecutive zeros and consecutive ones are bounded, and their frequency subsets , , and , , consisting… read more here.
Keywords: frequency sets; frequency; hausdorff dimension; univoque sequences ... See more keywords
Hausdorff dimension, heavy tails, and generalization in neural networks
Sign Up to like & getrecommendations! Published in 2020 at "Journal of Statistical Mechanics: Theory and Experiment"
DOI: 10.1088/1742-5468/ac3ae7
Abstract: Despite its success in a wide range of applications, characterizing the generalization properties of stochastic gradient descent (SGD) in non-convex deep learning problems is still an important challenge. While modeling the trajectories of SGD via… read more here.
Keywords: hausdorff dimension; neural networks; heavy tails; generalization ... See more keywords
A LOWER BOUND OF TOPOLOGICAL HAUSDORFF DIMENSION OF FRACTAL SQUARES
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DOI: 10.1142/s0218348x20501157
Abstract: Given an integer n ≥ 2 and a digit set ????⊂{0, 1,…,n − 1}2, there is a self-similar set K ⊂ ℝ2 satisfying the set equation K = (K + ????)/n. This set K is… read more here.
Keywords: topological hausdorff; hausdorff dimension; bound topological; lower bound ... See more keywords
Calculating Hausdorff Dimension in Higher Dimensional Spaces
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DOI: 10.3390/sym11040564
Abstract: In this paper, we prove the identity dim H(F) = d dim H(a?1(F)), where dim H denotesHausdorff dimension, F Rd, and a : [0, 1] ! [0, 1]d is a function whose constructive definition isaddressed… read more here.
Keywords: hausdorff dimension; higher dimensional; dimensional spaces; calculating hausdorff ... See more keywords