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Fractional dimension of sets in discrete spaces

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, , Citation M T Barlow and S J Taylor 1989 J. Phys. A: Math. Gen. 22 2621 DOI 10.1088/0305-4470/22/13/053

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1 Stat. Lab., Cambridge Univ., UK

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Abstract

The authors give a new definition dimH(A) for the dimension of an arbitrary subset of the lattice Bbb Zd. They establish elementary properties, and calculate the dimension for some examples. Finally, they announce a result which states that, if dimH(A)< d - 2, then A is transient for the simple random walk on Bbb Zd, and that if dimH(A) > d-2 then A is recurrent.

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