We prove that the Fourier dimension of any Boolean function with Fourier sparsity $s$ is at most $O\left(s^{ 2/3}\right)$. Our proof method yields an improved bound of $\widetilde{ O}(\sqrt{ s})$ assuming a conjecture of Tsang~\etal~\cite{ tsang}, that for every Boolean function of sparsity $s$ there is an affine subspace of $\mathbb{ F}_2^n$ of co-dimension $O(\poly\log s)$ restricted to which the function is constant. This conjectured bound is tight upto poly-logarithmic factors as the Fourier dimension and sparsity of the address function are quadratically separated. We obtain these bounds by observing that the Fourier dimension of a Boolean function is equivalent to its non-adaptive parity decision tree complexity, and then bounding the latter.
Categories: cs.CC