Linear-Size Constant-Depth Polylog-Treshold Circuits

We present a simple explicit construction giving unbounded fan-in circuits with $o(n)$ gates and depth $O(r)$ for the threshold function of $n$ variables when the threshold is at most $(log n)^r$, for any integer $r>0$. This improves a result of Atjai and Ben-Or, who showed the existence of circuits of size $n^{O(1)}$. This is the highest threshold for which polynomial-size, constant-depth circuits are possible.