Distances in sparse sets of large Hausdorff dimension

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.