The authors give a new definition dimH(A) for the dimension of an arbitrary subset of the lattice d. 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 d, and that if dimH(A) > d-2 then A is recurrent.
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Barlow et al. (1989) studied this question.
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