An Intermediate Value Property of Fractal Dimensions of Cartesian Product

Xiao-fang, Jiang; Liu, Qinghui; Zhou, Wen

doi:10.1142/s0218348x17500529

Cited by 5 publications

(1 citation statement)

References 6 publications

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“…When q = 0, the measures H q,t µ and P q,t µ do not depend on µ and they will be denoted by H t and P t respectively. The corresponding dimension inequalities for products of these measures are established in [23,31,16], the reader can be referred also to [20,33]. In this case (q = 0), these three inequalities are stated explicitly in [16,14,15,18].…”

Section: Introductionmentioning

confidence: 99%

A note on the generalized Hausdorff and packing measures of product sets in metric space

Guedri¹,

Attia²

2021

Preprint

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Let µ and ν be two Borel probability measures on two separable metric spaces X and Y respectively. For h, g be two Hausdorff functions and q ∈ R, we introduce and investigate the generalized pseudo-packing measure R q,h µ and the weighted generalized packing measure Q q,h µ to give some product inequalities :µ (E) P q,g ν (F ) for all E ⊆ X and F ⊆ Y, where H q,h µ and P q,h µ is the generalized Hausdorff and packing measures respectively. As an application, we prove that under appropriate geometric conditions, there exists a constant c such thatH q,h µ (E) P q,g ν (F ) ≤ c P q,hg µ (E × F ) P q,hg µ×ν (E × F ) ≤ c P q,h µ (E) P q,g ν (F ). These appropriate inequalities are more refined than well know results since we do no assumptions on µ, ν, h and g.

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