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 ( s ) . Our proof method yields an improved bound of Õ( √ s) assuming a conjecture of Tsang et. al. [TWXZ13], that for every Boolean function of sparsity s there is an affine subspace of F2 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.