  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.
