We prove that the Fourier dimension of any Boolean function with Fourier
sparsity s is at most O(s2/3). Our proof method yields an
improved bound of O(sβ) assuming a conjecture of
Tsang~\etal~\cite{tsang}, that for every Boolean function of sparsity s there
is an affine subspace of F2nβ 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