# Coin Theorems and the Fourier Expansion

## TL;DR

This paper explores the relationship between the complexity measures of Boolean functions in pseudorandomness, specifically comparing coin distinguishing ability and Fourier spectrum norms, revealing equivalences and implications for known bounds.

## Contribution

It establishes an equivalence between coin distinguishing advantage and Fourier coefficients for low bias coins, simplifying proofs of existing bounds and discussing potential converses for higher levels.

## Key findings

- Coin distinguishing advantage is proportional to the sum of level 1 Fourier coefficients for low bias.
- Known influence bounds follow from coin theorems via this equivalence.
- Discussion on the potential for converses at higher Fourier levels.

## 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 $\varepsilon = o(1/n)$, a function's distinguishing advantage in the coin problem is essentially equivalent to $\varepsilon$ 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 $\varepsilon$, and we discuss here the possibility of a converse.