Lower Bounds on Testing Functions of Low Fourier Degree

Abstract: We consider the problem of testing whether a Boolean function has Fourier degree $\leq k$ or it is $\epsilon$-far from any Boolean function with Fourier degree $\leq k$. We improve the known lower bound of $\Omega(k)$ \cite{BBM11,CGM10}, to $\Omega(k/\sqrt{\epsilon})$. The lower bound uses the recently discovered connections between property testing and communication complexity by Blais \textit{et. al.} \cite{BBM11}