  The distance set $\Delta(E)$ of a set $E$ consists of all non-negative
numbers that represent distances between pairs of points in $E$. This paper
studies sparse (less than full-dimensional) Borel sets in $\mathbb R^d$, $d
\geq 2$ with a focus on properties of their distance sets. Our results are of
four types. First, we generalize a classical result of Steinhaus (1920) to
Borel sets $E \subseteq [0,1]^d$ with $s$-dimensional Hausdorff content larger
than $(1 - \rho)$, for small $\rho > 0$ and $s$ close to $d$. For such sets, we
show that $\Delta(E) \supseteq [a, b]$, where $0<a<b$ depend only on $d$ and
$\rho$. This leads to our second result, a quantitative formulation of a
theorem of Mattila and Sj$\ddot{\text{o}}$lin (1999). For an arbitrary Borel
set $E \subseteq [0,1]^d$ of large Hausdorff dimension, we show that
$\Delta(E)$ contains a union of intervals whose lengths are dictated by cubes
where $E$ holds high density. This structure theorem in turn yields a tool for
identifying abundance of distances; this is the third contribution of this
article. It allows us to formulate a size property of a set that guarantees all
sufficiently large distances, generalizing earlier work of Bourgain (1986). It
can also be used to construct examples of totally disconnected sparse sets with
this property. Finally, we explore special features of $\Delta(E)$ if $E$ is
assumed to have certain structural regularity in addition to large Hausdorff
dimension. The additional regularity is harnessed via $L^2$-Fourier asymptotics
of measures supported on $E$. Applications of this phenomenon give new
information on $\Delta(E)$ when $E$ is locally uniformly $s$-dimensional and
quasi-regular, in the terminology of Strichartz (1990). A number of new
examples, counterexamples and open problems are discussed.
