  Thorup [FOCS'01, JACM'04] and Klein [SODA'01] independently showed that there
exists a $(1+\epsilon)$-approximate distance oracle for planar graphs with $O(n
(\log n)\epsilon^{-1})$ space and $O(\epsilon^{-1})$ query time. While the
dependency on $n$ is nearly linear, the space-query product of their oracles
depend quadratically on $1/\epsilon$. Many follow-up results either improved
the space \emph{or} the query time of the oracles while having the same,
sometimes worst, dependency on $1/\epsilon$. Kawarabayashi, Sommer, and Thorup
[SODA'13] were the first to improve the dependency on $1/\epsilon$ from
quadratic to nearly linear (at the cost of $\log^*(n)$ factors). It is
plausible to conjecture that the linear dependency on $1/\epsilon$ is optimal:
for many known distance-related problems in planar graphs, it was proved that
the dependency on $1/\epsilon$ is at least linear.
  In this work, we disprove this conjecture by reducing the dependency of the
space-query product on $1/\epsilon$ from linear all the way down to
\emph{subpolynomial} $(1/\epsilon)^{o(1)}$. More precisely, we construct an
oracle with $O(n\log(n)(\epsilon^{-o(1)} + \log^*n))$ space and
$\log^{2+o(1)}(1/\epsilon)$ query time. Our construction is the culmination of
several different ideas developed over the past two decades.
