On the Distribution of the Fourier Spectrum of Halfspaces
  Bourgain showed that any noise stable Boolean function $f$ can be well-approximated by a junta. In this note we give an exponential sharpening of the parameters of Bourgain's result under the additional assumption that $f$ is a halfspace. 
Ilias Diakonikolas and Ragesh Jaiswal and Rocco A. Servedio and   Li-Yang Tan and Andrew Wan
cs.CC cs.DM math.PR