Small set high dimensional expansion over general sheaves has recently became central to several major breakthroughs. These include the construction of classical locally testable and quantum LDPC codes with constant rate and linear distance, $\Omega(n)$-levels lower bounds for the Sum-of-Squares hierarchy, and the proof of the NLTS conjecture.

In this work, we improve upon the state-of-the-art results for small set expansion in simplicial complexes, for both constant and general sheaves. Compared to one line of work (e.g., [KM22, DD24]), we improve upon their result by obtaining \emph{strong expansion} for small sets. Compared to another line of work (e.g., [KKL14, EK16, KM21, FK22]), we get an \emph{exponential improvement} in the size of sets for which expansion is guaranteed.

Our result is based on a novel technique that bridges these two lines of work via a ``local vs. global'' case analysis: for ``local'' sets, which are concentrated within small links, we adapt ideas from the ``fat machinery'' of [KKL14, EK16]; for ``global'' sets, which are spread across the complex, we exploit global averaging operators in the spirit of [KM22, DD24].

Small set expansion in high dimensional expanders is of great importance, e.g., towards proving cosystolic expansion, local testability of codes and constructions of good quantum codes.

In this work we improve upon the state of the art results of small set expansion in high dimensional expanders. Our improvement is either on the expansion quality or on the size of sets for which expansion is guaranteed.

One line of previous works [KM22, DD24] has obtained weak expansion for small sets, which is sufficient for deducing cosystolic expansion of one dimension below. We improve upon their result by showing strong expansion for small sets.

Another line of works [KKL14, EK16, KM21] has shown strong expansion for small sets. However, they obtain it only for very small sets. We get an exponential improvement on the size of sets for which expansion is guaranteed by these prior works.

Interestingly, our result is obtained by bridging between these two lines of works. The works of [KM22, DD24] use global averaging operators in order to obtain expansion for larger sets. However, their method could be utilized only on sets that are cocycle-like. We show how to combine these global averaging operators with ideas from the so-called ``fat machinery'' of [KKL14, EK16, KM21] in order to apply them for general sets.