Sub-linear Upper Bounds on Fourier dimension of Boolean Functions in terms of Fourier sparsity

We prove that the Fourier dimension of any Boolean function with Fourier sparsity $s$ is at most $O\left(s^{2\mathrm{/}3}\right)$. Our proof method yields an improved bound of $\tilde{O}(\sqrt{s})$ 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 $O(\text{poly}\log s)$ 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.