  Cut and spectral sparsification of graphs have numerous applications,
including e.g. speeding up algorithms for cuts and Laplacian solvers. These
powerful notions have recently been extended to hypergraphs, which are much
richer and may offer new applications. However, the current bounds on the size
of hypergraph sparsifiers are not as tight as the corresponding bounds for
graphs.
  Our first result is a polynomial-time algorithm that, given a hypergraph on
$n$ vertices with maximum hyperedge size $r$, outputs an $\epsilon$-spectral
sparsifier with $O^*(nr)$ hyperedges, where $O^*$ suppresses $(\epsilon^{-1}
\log n)^{O(1)}$ factors. This size bound improves the two previous bounds:
$O^*(n^3)$ [Soma and Yoshida, SODA'19] and $O^*(nr^3)$ [Bansal, Svensson and
Trevisan, FOCS'19]. Our main technical tool is a new method for proving
concentration of the nonlinear analogue of the quadratic form of the Laplacians
for hypergraph expanders.
  We complement this with lower bounds on the bit complexity of any compression
scheme that $(1+\epsilon)$-approximates all the cuts in a given hypergraph, and
hence also on the bit complexity of every $\epsilon$-cut/spectral sparsifier.
These lower bounds are based on Ruzsa-Szemer\'edi graphs, and a particular
instantiation yields an $\Omega(nr)$ lower bound on the bit complexity even for
fixed constant $\epsilon$. This is tight up to polylogarithmic factors in $n$,
due to recent hypergraph cut sparsifiers of [Chen, Khanna and Nagda, FOCS'20].
  Finally, for directed hypergraphs, we present an algorithm that computes an
$\epsilon$-spectral sparsifier with $O^*(n^2r^3)$ hyperarcs, where $r$ is the
maximum size of a hyperarc. For small $r$, this improves over $O^*(n^3)$ known
from [Soma and Yoshida, SODA'19], and is getting close to the trivial lower
bound of $\Omega(n^2)$ hyperarcs.
