Bounded Independence Fools Halfspaces

We show that any distribution on {-1,+1}ⁿ that is k-wise independent fools any halfspace (a.k.a. threshold) h : {-1,+1}ⁿ ¿ {-1,+1}, i.e., any function of the form h(x) = sign(¿_i=1 ⁿ w_iX_i - ¿) where the w₁,..., w_n, ¿ are arbitrary real numbers, with error ¿ for k = O(¿^-2 log²(1/¿)). Our result is tight up to log(1/¿) factors. Using standard constructions of k-wise independent distributions, we obtain the first explicit pseudorandom generators G : {-1,+1}^s ¿ {-1,+1}ⁿ that fool halfspaces. Specifically, we fool halfspaces with error e and seed length s = k · log n = O(log n · ¿^-2 log²(1/¿)). Our approach combines classical tools from real approximation theory with structural results on halfspaces by Servedio (Comput. Complexity 2007).