  We investigate the relation between $\delta$ and $\epsilon$ required for
obtaining a $(1+\delta)$-approximation in time $N^{2-\epsilon}$ for closest
pair problems under various distance metrics, and for other related problems in
fine-grained complexity.
  Specifically, our main result shows that if it is impossible to (exactly)
solve the (bichromatic) inner product (IP) problem for vectors of dimension $c
\log N$ in time $N^{2-\epsilon}$, then there is no $(1+\delta)$-approximation
algorithm for (bichromatic) Euclidean Closest Pair running in time
$N^{2-2\epsilon}$, where $\delta \approx (\epsilon/c)^2$ (where $\approx$ hides
$\polylog$ factors). This improves on the prior result due to Chen and Williams
(SODA 2019) which gave a smaller polynomial dependence of $\delta$ on
$\epsilon$, on the order of $\delta \approx (\epsilon/c)^6$. Our result implies
in turn that no $(1+\delta)$-approximation algorithm exists for Euclidean
closest pair for $\delta \approx \epsilon^4$, unless an algorithmic improvement
for IP is obtained. This in turn is very close to the approximation guarantee
of $\delta \approx \epsilon^3$ for Euclidean closest pair, given by the best
known algorithm of Almam, Chan, and Williams (FOCS 2016). By known reductions,
a similar result follows for a host of other related problems in fine-grained
hardness of approximation.
  Our reduction combines the hardness of approximation framework of Chen and
Williams, together with an MA communication protocol for IP over a small
alphabet, that is inspired by the MA protocol of Chen (Theory of Computing,
2020).
