Approximate Degree, Weight, and Indistinguishability

We prove that the OR function on {-1, 1}(n) can be pointwise approximated with error epsilon by a polynomial of degree O(k) and weight 2(O(n log(1)(/epsilon)()/k)), for any k >= root n log(1/epsilon). This result is tight for any k <= (1 - Omega(1))n. Previous results were either not tight or had epsilon = Omega(1). In general, we obtain a tight approximate degree-weight result for any symmetric function. Building on this, we also obtain an approximate degree-weight result for bounded-width CNF. For these two classes no such result was known. One motivation for such results comes from the study of indistinguishability. Two distributions P, Q over n-bit strings are (k, delta)-indistinguishable if their projections on any k bits have statistical distance at most delta. The above approximations give values of (k, delta) that suffice to fool OR, symmetric functions, and boundedwidth CNF, and the first result is tight for all k while the second result is tight for k <= (1 - Omega(1))n. We also show that any two (k, delta)-indistinguishable distributions are O(n(k/2)delta)-close to two distributions that are (k, 0)-indistinguishable, improving the previous bound of O(n)(k)delta. Finally, we present proofs of some known approximate degree lower bounds in the language of indistinguishability, which we find more intuitive.