Coin Theorems and the Fourier Expansion
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 constantwidth readonce branching programs) in fact follow as a blackbox from the corresponding coin theorems, thereby simplifying the proofs of some known results in the literature. For higher levels, it is wellknown 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2019
 DOI:
 10.48550/arXiv.1906.03743
 arXiv:
 arXiv:1906.03743
 Bibcode:
 2019arXiv190603743A
 Keywords:

 Computer Science  Computational Complexity
 EPrint:
 15 pages, 1 figure