# Advice lower bounds for the dense model theorem

## Abstract

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 D that is δ-dense in a distribution that is ε -indistinguishable from uniform, there exists a "dense model" for D, that is, a distribution that is δ-dense in the uniform distribution and is ε′-indistinguishable from D. This ε -indistinguishability is with respect to an arbitrary small class of functions F. For the natural case where ε ≥ Δ (ε δ) and ε ≥ δO(1), our lower bound implies that Δ( √ (1/ε) log(1/δ) log |F| ) 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 O ((1/ε2) log(1/δ) log |F|). 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.

## Publication Title

ACM Transactions on Computation Theory

## Recommended Citation

Watson, T.
(2014). Advice lower bounds for the dense model theorem.* ACM Transactions on Computation Theory**, 7* (1)
https://doi.org/10.1145/2676659