Let f:{-1,1}^n -> R be a real function on the hypercube, given by its
discrete Fourier expansion, or, equivalently, represented as a multilinear
polynomial. We say that it is Boolean if its image is in {-1,1}.
We show that every function on the hypercube with a sparse Fourier expansion
must either be Boolean or far from Boolean. In particular, we show that a
multilinear polynomial with at most k terms must either be Boolean, or output
values different than -1 or 1 for a fraction of at least 2/(k+2)^2 of its
domain.
It follows that given oracle access to f, together with the guarantee that
its representation as a multilinear polynomial has at most k terms, one can
test Booleanity using O(k^2) queries. We show an \Omega(k) queries lower bound
for this problem.
Our proof crucially uses Hirschman's entropic version of Heisenberg's
uncertainty principle.Comment: 15 page
