  The FOCS'19 paper of Le and Solomon, culminating a long line of research on
Euclidean spanners, proves that the lightness (normalized weight) of the greedy
$(1+\epsilon)$-spanner in $\mathbb{R}^d$ is $\tilde{O}(\epsilon^{-d})$ for any
$d = O(1)$ and any $\epsilon = \Omega(n^{-\frac{1}{d-1}})$ (where $\tilde{O}$
hides polylogarithmic factors of $\frac{1}{\epsilon}$), and also shows the
existence of point sets in $\mathbb{R}^d$ for which any $(1+\epsilon)$-spanner
must have lightness $\Omega(\epsilon^{-d})$. Given this tight bound on the
lightness, a natural arising question is whether a better lightness bound can
be achieved using Steiner points.
  Our first result is a construction of Steiner spanners in $\mathbb{R}^2$ with
lightness $O(\epsilon^{-1} \log \Delta)$, where $\Delta$ is the spread of the
point set. In the regime of $\Delta \ll 2^{1/\epsilon}$, this provides an
improvement over the lightness bound of Le and Solomon [FOCS 2019]; this regime
of parameters is of practical interest, as point sets arising in real-life
applications (e.g., for various random distributions) have polynomially bounded
spread, while in spanner applications $\epsilon$ often controls the precision,
and it sometimes needs to be much smaller than $O(1/\log n)$. Moreover, for
spread polynomially bounded in $1/\epsilon$, this upper bound provides a
quadratic improvement over the non-Steiner bound of Le and Solomon [FOCS 2019],
We then demonstrate that such a light spanner can be constructed in
$O_{\epsilon}(n)$ time for polynomially bounded spread, where $O_{\epsilon}$
hides a factor of $\mathrm{poly}(\frac{1}{\epsilon})$. Finally, we extend the
construction to higher dimensions, proving a lightness upper bound of
$\tilde{O}(\epsilon^{-(d+1)/2} + \epsilon^{-2}\log \Delta)$ for any $3\leq d =
O(1)$ and any $\epsilon = \Omega(n^{-\frac{1}{d-1}})$.
