The discrete torus is the graph on Zₖⁿ = ( Z/kZ )ⁿ in which x = (xᵢ )₁ⁿ is joined to y = (yᵢ )₁ⁿ if for some i there is xᵢ = yᵢ ± 1 and xⱼ = yⱼ for all j ≠ i. For a set A ⊂ Zₖⁿ and a natural number t, let A( t ) be the set of vertices of Zₖⁿ within distance t of A. The main aim of this paper is to give a best possible lower bound for | A₍ₜ₎ | in terms of $| A |$, for even values of k.
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Bollobás et al. (1990) studied this question.
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