COS 598 Week 8 Degree / discrepancy Theorem . Group representations

1 Degree/discrepancy theorem The ideas presented in this section are due to [She07]. The heuristic idea of the construction is as follows. We start with some Boolean function f : {−1, 1} n → {−1, 1} of " high degree " We then produce a pair " Nice " distributionµ on {−1, 1} n " Useful " matrix M such that disc µ (M) is low. This can turn out to be useful in proving lower bounds. To begin with, we need the following Definition 1.1 (Threshold degree). We will say that the threshold degree of f is at most d (and write thr(f) ≤ d) if there exists a degree d polynomial P (with real coefficients, but can in fact assume integral) such that ∀x ∈ {−1, 1} n , f (x) = sgn(P (x)). With this definition, we are ready to state the first ingredient of the construction. Theorem 1.2. If thr(f) = d then there exists a distribution µ such that ∀P ∈ R[x], with deg P < d, E µ (f · P) = 0 Sketch. We shall make use of Farkas' Lemma, a very nice discussion of which is provided in [Tao]. The statement of the lemma is simple: If we have a system of linear inequalities P i (x) ≥ 0, then either it has a solution, or it implies a 1