Juntas In The (1)-Grid And Lipschitz Maps Between Discrete Tori

We show that if A< subset of>[k]n, then A is E-close to a junta depending upon at most exp(O(|A|/(kn-1E))) coordinates, where A denotes the edge-boundary of A in the 1-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 1-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. (c) 2015 Wiley Periodicals, Inc. Random Struct. Alg., 49, 253-279, 2016