Coin Theorems and the Fourier Expansion
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In this note we compare two measures of the complexity of a class F of Boolean functions studied in (unconditional) pseudorandomness: 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 F (the Fourier growth). We show that for coins with low bias ε = o(1/n), a function's distinguishing advantage in the coin problem is essentially equivalent to ε 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 ε, and we discuss here the possibility of a converse.
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