• Bulletin of the Korean Mathematical Society
• /
• v.52 no.5
• /
• pp.1489-1493
• /
• 2015

For a subset $E{\subseteq}\mathbb{R}^d$ and $x{\in}\mathbb{R}^d$, the local Hausdorff dimension function of E at x and the local packing dimension function of E at x are defined by $$dim_{H,loc}(x,E)=\lim_{r{\searrow}0}dim_H(E{\cap}B(x,r))$$, $$dim_{P,loc}(x,E)=\lim_{r{\searrow}0}dim_P(E{\cap}B(x,r))$$, where $dim_H$ and $dim_P$ denote the Hausdorff dimension and the packing dimension, respectively. In this note we give a short and simple proof showing that for any pair of continuous functions $f,g:\mathbb{R}^d{\rightarrow}[0,d]$ with $f{\leq}g$, it is possible to choose a set E that simultaneously has f as its local Hausdorff dimension function and g as its local packing dimension function.

• Bulletin of the Korean Mathematical Society
• /
• v.41 no.1
• /
• pp.13-18
• /
• 2004

We defined Cantor dimensions of a perturbed Cantor set, and investigated a relation between these dimensions and Hausdorff and packing dimensions of a perturbed Cantor set. In this paper, we introduce another expressions of the Cantor dimensions. Using these, we study some informations which can be derived from power equations induced from contraction ratios of a perturbed Cantor set to give its Hausdorff or packing dimension. This application to a deranged Cantor set gives us an estimation of its Hausdorff and packing dimensions, which is a generalization of the Cantor dimension theorem.

• Bulletin of the Korean Mathematical Society
• /
• v.35 no.3
• /
• pp.397-408
• /
• 1998

We often use the Hausdorff dimension as a tool of measuring how complicate the fractal is. But it is usually very difficult to calculate that value. So there have been many tries to find the dimension of the given set and most of these are related to the density theorem of invariant measure. The aims of this paper are to introduce the k-irreducible subsimilar sets as a generalization of the set defined by V.Drobot and J.Turner in ([1]) and calculate their Hausdorff dimensions by using algebraic methods.

• Journal of the Chungcheong Mathematical Society
• /
• v.21 no.2
• /
• pp.157-163
• /
• 2008

We study essentially disjoint one dimensionally indexed classes whose members are distribution sets of a self-similar Cantor set. The Hausdorff dimension of the union of distribution sets in a same class does not increases the Hausdorff dimension of the characteristic distribution set in the class. Further we study the Hausdorff dimension of some uncountable union of distribution sets.

• Bulletin of the Korean Mathematical Society
• /
• v.54 no.4
• /
• pp.1173-1183
• /
• 2017

We give some sufficient conditions for the null and infinite derivatives of the $Riesz-N{\acute{a}}gy-Tak{\acute{a}}cs$ (RNT) singular function. Using these conditions, we show that the Hausdorff dimension of the set of the infinite derivative points of the RNT singular function coincides with its packing dimension which is positive and less than 1 while the Hausdorff dimension of the non-differentiability set of the RNT singular function does not coincide with its packing dimension 1.

• Journal of the Korean Mathematical Society
• /
• v.31 no.4
• /
• pp.539-544
• /
• 1994

We [1] investigated the Hausdorff dimension and the packing dimension of a certain perturbed Cantor set whose ratios are unformly bounded. In this paper, we consider a specific Cantor set whose ratios are not necessarily uniformly bounded but satisfy some other conditions. In fact, in the hypothesis, only the condition of the unform boundedness of ratios on the set is substituted by a "*-condition". We use energy theory related to Hausdorff dimension in this study while we [1] used Hausdorff density theorem to find the Hausdorff dimension of some perturbed Cantor set. In the end, we given an example which explains aformentioned facts.ned facts.

• Communications of the Korean Mathematical Society
• /
• v.19 no.1
• /
• pp.113-117
• /
• 2004

A deranged Cantor set (without the uniform bounded-ness condition away from zero of contraction ratios) whose weak local dimensions for all points coincide has its Hausdorff dimension of the same value of weak local dimension. We will show it using an energy theory instead of Frostman's density lemma which was used for the case of the deranged Cantor set with the uniform boundedness condition of contraction ratios. In the end, we will give an example of such a deranged Cantor set.

• Journal of the Chungcheong Mathematical Society
• /
• v.28 no.1
• /
• pp.29-38
• /
• 2015

We compute the Hausdorff and packing dimensions of the cylindrical lower or upper local dimension set for a self-similar measure having different minimum and maximum of the local dimension on a self-similar set satisfying the open set condition.

• Journal of the Korean Mathematical Society
• /
• v.41 no.5
• /
• pp.933-944
• /
• 2004

Packing dimension of a set is an upper bound for the packing dimensions of measures on the set. Recently the packing dimension of statistically self-similar Cantor set, which has uniform distributions for contraction ratios, was shown to be its Hausdorff dimension. We study the method to find an upper bound of packing dimensions and the upper Renyi dimensions of measures on a statistically quasi-self-similar Cantor set (its packing dimension is still unknown) which has non-uniform distributions of contraction ratios. As results, in some statistically quasi-self-similar Cantor set we show that every probability measure on it has its subset of full measure whose packing dimension is also its Hausdorff dimension almost surely and it has its subset of full measure whose packing dimension is also its Hausdorff dimension almost surely for almost all probability measure on it.

• Journal of applied mathematics & informatics
• /
• v.14 no.1_2
• /
• pp.397-403
• /
• 2004

Cutler showed some duality results about Hausdorff and packing dimensions of a measure on a compact set in Euclidean space if its s-dimensional Hausdorff measure or packing measure is positive. We show that the similar results in a perturbed Cantor set hold according to its quasi s-dimensional Hausdorff measure or packing measure and we find concrete measures in this case while Cutler showed the existence of such measures. Finally under some strong condition, we give a concrete measure whose Hausdorff and packing dimensions are the same as those of the perturbed Cantor set without the condition that it has positive s-dimensional Hausdorff or packing measures.