  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.
