Exponential lower bounds for depth three Boolean circuits

We consider the class Sigma (k)(3) of unbounded fan-in depth three Boolean circuits, for which the bottom fan-in is limited by k and the top gate is a n OR. It is known that the smallest such circuit computing the parity funct ion has Ohm>(*) over bar * (2(epsilonn/k)) gates (for k = O(n(1/2))) for so me epsilon > 0, and this was the best lower bound known for explicit (P-tim e computable) functions. In this paper, for k = 2, we exhibit functions in uniform NC1 that require 2(n-o(n)) size depth 3 circuits. The main tool is a theorem that shows that any Sigma (2)(3) circuit on n variables that acce pts a inputs and has size s must be constant on a projection (subset define d by equations of the form x(i) = 0, x(i) = 1, x(i) = x(j) or x(i) = (x) ov er bar (j)) of dimension at least log(a/s)/log n.