Advice Lower Bounds for the Dense Model Theorem

We prove a lower bound on the amount of nonuniform advice needed by black-box reductions for the Dense Model Theorem of Green, Tao, and Ziegler, and of Reingold, Trevisan, Tulsiani, and Vadhan. The latter theorem roughly says that for every distribution <i>D</i> that is Δ-dense in a distribution that is <i>ε</i>′-indistinguishable from uniform, there exists a “dense model” for <i>D</i>, that is, a distribution that is <i>Δ</i>-dense in the uniform distribution and is <i>ε</i>-indistinguishable from <i>D</i>. This <i>ε</i>-indistinguishability is with respect to an arbitrary small class of functions <i>F</i>. For the natural case where <i>ε</i>′ ≥ Ω(<i>εΔ</i>) and <i>ε</i> ≥ <i>Δ</i><i>O</i>(1), our lower bound implies that Ω(√(1/<i>ε</i>) log(1/<i>Δ</i>)·log|<i>F</i>|) advice bits are necessary for a certain type of reduction that establishes a stronger form of the Dense Model Theorem (and which encompasses all known proofs of the Dense Model Theorem in the literature). There is only a polynomial gap between our lower bound and the best upper bound for this case (due to Zhang), which is <i>O</i>((1/<i>ε</i>2)log(1/<i>Δ</i>)·log|<i>F</i>|). Our lower bound can be viewed as an analogue of list size lower bounds for list-decoding of error-correcting codes, but for “dense model decoding” instead.