# On the Fourier spectrum of functions on Boolean cubes

## TL;DR

This paper establishes a bound on the sum of Fourier coefficients of Boolean functions of degree d, revealing new techniques and connections to quantum information theory, and highlighting differences from complex polynomial cases.

## Contribution

The paper proves a novel bound on Fourier coefficients of Boolean functions, introducing techniques that differ from those used in complex or real polynomial analysis.

## Key findings

- Bound on the ll_{2d/(d+1)}-sum of Fourier coefficients
- New techniques for Boolean functions analysis
- Connections to quantum information theory

## Abstract

Let $f$ be a real-valued, degree-$d$ Boolean function defined on the $n$-dimensional Boolean cube $\{\pm 1\}^{n}$, and $f(x) = \sum_{S \subset \{1,\ldots,d\}} \widehat{f}(S) \prod_{k \in S} x_k$ its Fourier-Walsh expansion. The main result states that there is an absolute constant $C >0$ such that the $\ell_{2d/(d+1)}$-sum of the Fourier coefficients of $f:\{\pm 1\}^{n} \rightarrow [-1,1]$ is bounded by $\leq C^{\sqrt{d \log d}}$. It was recently proved that a similar result holds for complex-valued polynomials on the $n$-dimensional poly torus $\mathbb{T}^n$, but that in contrast to this a replacement of the $n$-dimensional torus $\mathbb{T}^n$ by $n$-dimensional cube $[-1, 1]^n$ leads to a substantially weaker estimate. This in the Boolean case forces us to invent novel techniques which differ from the ones used in the complex or real case. We indicate how our result is linked with several questions in quantum information theory.