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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 is at most . Our proof method yields an
improved bound of assuming a conjecture of
Tsang~\etal~\cite{tsang}, that for every Boolean function of sparsity there
is an affine subspace of 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
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