We show that if A ⊂ [k]n, then A is ε -close to a junta depending upon at most exp(O(|∂A|/(kn-1ε))) coordinates, where ∂A denotes the edge-boundary of A in the l1 -grid. This bound is sharp up to the value of the absolute constant in the exponent. This result can be seen as a generalisation of the Junta theorem for the discrete cube, from , or as a characterisation of large subsets of the l1 -grid whose edge-boundary is small. We use it to prove a result on the structure of Lipschitz functions between two discrete tori; this can be seen as a discrete, quantitative analogue of a recent result of Austin . We also prove a refined version of our junta theorem, which is sharp in a wider range of cases.
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- Boolean functions