Bohr's phenomenon for functions on the Boolean cube

We study the asymptotic decay of the Fourier spectrum of real functions f:{−1,1}N⟶R in the spirit of Bohr's phenomenon from complex analysis. Every such function admits a canonical representation through its Fourier-Walsh expansion f(x)=∑S⊂{1,…,N}fˆ(S)xS, where xS=∏k∈Sxk. Given a class F of functions on the Boolean cube {−1,1}N, the Boolean radius of F is defined to be the largest ρ≥0 such that ∑S|fˆ(S)|ρ|S|≤‖f‖∞ for every f∈F. We give the precise asymptotic behavior of the Boolean radius of several natural subclasses of functions on finite Boolean cubes, as e.g. the class of all real functions on {−1,1}N, the subclass made of all homogeneous functions or certain threshold functions. Compared with the classical complex situation subtle differences as well as striking parallels occur.