Juntas in the ℓ₁-grid and Lipschitz maps between discrete tori

We show that if A ⊂ [k]ⁿ, then A is ε -close to a junta depending upon at most exp(O(|∂A|/(k^n-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 [6], 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 [1]. We also prove a refined version of our junta theorem, which is sharp in a wider range of cases.