Boolean Functions with small Spectral Norm

Abstract.

Let $f : {\mathbb{F}}^{n}_{2} \rightarrow \{0, 1\}$ be a boolean function, and suppose that the spectral norm $\|f\|_{A} := \sum_{r} \mid \widehat{f}(r)\mid$ of f is at most M. Then $\mathop {f = \sum\limits^{L}_{j=1}\pm 1_{{H}_{j}}},$ where $L \leq 2^{{{2}^{CM}}^{4}}$ and each H _j is a subgroup of ${\mathbb{F}}^{n}_{2}$ . This result may be regarded as a quantitative analogue of the Cohen-Helson-Rudin structure theorem for idempotent measures in locally compact abelian groups.