Double Balanced Sets in High Dimensional Expanders

TL;DR

This paper introduces 'double balanced sets' as a new pseudorandomness concept in high-dimensional expanders, demonstrating their strong expansion properties and applications to cosystolic expanders, extending analysis beyond Fourier methods.

Contribution

It proposes the novel concept of double balanced sets and establishes their strong expansion properties in one-sided local spectral expanders, broadening the scope of pseudorandom set analysis.

Findings

Small double balanced sets exhibit strong expansion properties.

Cohomologies in cosystolic expanders are double balanced.

Improved lower bounds on minimal distance of cosystolic expanders.

Abstract

Recent works have shown that expansion of pseudorandom sets is of great importance. However, all current works on pseudorandom sets are limited only to product (or approximate product) spaces, where Fourier Analysis methods could be applied. In this work we ask the natural question whether pseudorandom sets are relevant in domains where Fourier Analysis methods cannot be applied, e.g., one-sided local spectral expanders. We take the first step in the path of answering this question. We put forward a new definition for pseudorandom sets, which we call ``double balanced sets''. We demonstrate the strength of our new definition by showing that small double balanced sets in one-sided local spectral expanders have very strong expansion properties, such as unique-neighbor-like expansion. We further show that cohomologies in cosystolic expanders are double balanced, and use the newly derived strong expansion properties of double balanced sets in order to obtain an exponential improvement over the current state of the art lower bound on their minimal distance.