Boolean Functions with small Spectral Norm

Abstract

Abstract. Let f : F n 2 → {0, 1} be a boolean function, and suppose that the spectral normand each H j is a subgroup of F n 2 .This result may be regarded as a quantitative analogue of the Cohen-Helson-Rudin structure theorem for idempotent measures in locally compact abelian groups.

A quantitative version of the idempotent theorem in harmonic analysis

“…As in our earlier paper [12], it is not possible to effect such a procedure entirely within the "category" of Z-valued functions. One must consider, more generally, functions which are ε-almost Z-valued, that is to say take values in Z + [−ε, ε].…”

“…A result of this type in the finite field setting, where S is just a subgroup system in F n 2 , was obtained in [12]. The argument there, which was a combination of [12,Lemma 3.4] and [12,Prop. 3.7], was somewhat elaborate and involved polynomials which are small near small integers.…”

“…The argument we give here is different and is close to the main argument in [10] (in fact, it is very close to the somewhat simpler argument, leading to a bound of O(log −1/4 p), sketched just after Lemma 4.1 of that paper). In the finite field setting it is simpler than that given in [12,Sec. 3] …”

“…In our earlier work [12] we used (a refinement of) Ruzsa's analogue of Freiman's theorem, which gives a fairly strong characterisation of subsets A ⊆ F n 2 satisfying a small doubling condition |A + A| K|A|. An analogue of this theorem for any abelian group was obtained in [11].…”

“…The corresponding argument in [12] (which comes near the end of Proposition 5.1) is rather short, but in the setting of a general Bourgain system we must work a little harder.…”