### Volume 2020

#### Coin Theorems and the Fourier Expansion

Rohit Agrawal
Department of Computer Science
Harvard University
Cambridge, MA, USA
rohitagr AT seas DOT harvard DOT edu

August 31, 2020

#### Abstract

In this note we compare two measures of the complexity of a class $\mathcal{F}$ of Boolean functions studied in (unconditional) pseudorandomness: $\mathcal{F}$'s ability to distinguish between biased and uniform coins (the coin problem), and the norms of the different levels of the Fourier expansion of functions in $\mathcal{F}$ (the Fourier growth). We show that for coins with low bias $\epsilon = o(1/n)$, a function's distinguishing advantage in the coin problem is essentially equivalent to $\epsilon$ times the sum of its level $1$ Fourier coefficients, which in particular shows that known level $1$ and total influence bounds for some classes of interest (such as constant-width read-once branching programs) in fact follow as a black-box from the corresponding coin theorems, thereby simplifying the proofs of some known results in the literature. For higher levels, it is well-known that Fourier growth bounds on all levels of the Fourier spectrum imply coin theorems, even for large $\epsilon$, and we discuss here the possibility of a converse.

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