From expanders to hitting distributions and simulation theorems

Recently, Chattopadhyay et al. (\cite{chattopadhyay2017simulation}) proved that any gadget having so called \emph{hitting distributions} admits deterministic "query-to-communication" simulation theorem. They applied this result to Inner Product, Gap Hamming Distance and Indexing Function. They also demonstrated that previous works used hitting distributions implicitly (\cite{goos2015deterministic} for Indexing Function and \cite{wu2017raz} for Inner Product). In this paper we show that any expander in which any two distinct vertices have at most one common neighbor can be transformed into a gadget possessing good hitting distributions. We demonstrate that this result is applicable to affine plane expanders and to Lubotzky-Phillips-Sarnak construction of Ramanujan graphs . In particular, from affine plane expanders we extract a gadget achieving the best known trade-off between the arity of outer function and the size of gadget. More specifically, when this gadget has $k$ bits on input, it admits a simulation theorem for all outer function of arity roughly $2^{k/2}$ or less (the same was also known for $k$-bit Inner Product, (\cite{chattopadhyay2017simulation})). In addition we show that, unlike Inner Product, underlying hitting distributions in our new gadget are "polynomial-time listable" in the sense that their supports can be written down in time $2^{O(k)}$, i.e, in time polynomial in size of gadget's matrix.