A Slight Sharpening of LMN

Linial, Mansour and Nisan [7] proved that a function computed by a small-depth circuit of limited size has most of its Fourier support on small sets. We improve their bounds. When the bottom fanin is bounded we use essentially their argument, but to reduce the general case to this case without a loss in the asymptotic bounds requires a new argument. Warning: Essentially this paper is accepted for publication in Journal of Computer and System Sciences and is subject to copyright restrictions. In particular it is for personal use only.