Near-Optimal Upper Bound on Fourier Dimension of Boolean Functions in Terms of Fourier Sparsity

Abstract

We prove that the Fourier dimension of any Boolean function with Fourier sparsity $s$ is at most $O\left( \sqrt{s} \log s\right)$. This bound is tight up to a factor of $O(\log s)$ as the Fourier dimension and sparsity of the addressing function are quadratically related. We obtain our result by bounding the non-adaptive parity decision tree complexity, which is known to be equivalent to the Fourier dimension. A consequence of our result is that XOR functions have a one way deterministic communication protocol of communication complexity $O(\sqrt{r} \log r)$, where $r$ is the rank of its communication matrix.