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arXiv · 1007.4222

Strict inequality in the box-counting dimension product formulas

Nick Sharples
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Abstract

It is known that the upper box-counting dimension of a Cartesian product satisfies the inequality $\dim_{B}\left(F\times G\right)\leq \dim_{B}\left(F\right) + \dim_{B}\left(G\right)$ whilst the lower box-counting dimension satisfies the inequality $\dim_{LB}\left(F\times G\right)\geq \dim_{LB}\left(F\right) + \dim_{LB}\left(G\right)$. We construct Cantor-like sets to demonstrate that both of these inequalities can be strict.

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Nick Sharples. 2010-07-23. Strict inequality in the box-counting dimension product formulas. https://arxiv.org/abs/1007.4222

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