Bohr's phenomenon for functions on the Boolean cube

We study the asymptotic decay of the Fourier spectrum of real functions $f\text{\hspace{0.17em}⁣}:\{-1,1{\}}^N\to \mathbb{R}$ in the spirit of Bohr’s phenomenon from complex analysis. Every such function admits a canonical representation through its Fourier-Walsh expansion $f(x) = \sum_{S\subset {1,\ldots,N}}\widehat{f}(S) x^S ,,$where$x^S = \prod_{k \in S} x_k$. Given a class $\mathcal{F}$ of functions on the Boolean cube $\{-1,1{\}}^N$, the Boolean radius of $\mathcal{F}$ is defined to be the largest $\rho \ge 0$ such that ${\sum }_S\mathrm{\mid }\widehat{f}(S)\mathrm{\mid }{\rho }^{\mathrm{\mid }S\mathrm{\mid }}\le \mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}$ for every $f \in $\mathcal{F}$$. We give the precise asymptotic behaviour 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.