We show that if Aβ[k]n, then A is Ο΅-close to a junta
depending upon at most exp(O(β£βAβ£/(knβ1Ο΅))) coordinates,
where βA denotes the edge-boundary of A in the β1-grid. This
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 [E. Friedgut, Boolean functions with low average sensitivity depend on few
coordinates, Combinatorica 18 (1998), 27-35], or as a characterization of large
subsets of the β1-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
[T. Austin, On the failure of concentration for the ββ-ball,
preprint]. We also prove a refined version of our junta theorem, which is sharp
in a wider range of cases.Comment: 29 pages. A mistake in Example 2 (pointed out by an anonymous
referee) has now been correcte