Are most Boolean functions determined by low frequencies?

Abstract

We ask whether most Boolean functions are determined by their low frequencies. We show a partial result: for almost every function $f:\{-1,1{\}}^p\to \{-1,1\}$ there exists a function $f^{\prime }:\{-1,1{\}}^p\to (-1,1)$ that has the same frequencies as $f$ up to dimension $(1/2-o(1))p$.

10 pages. A couple of minor inaccuracies fixed