Spherical averaging operators are ubiquitous, appearing in solutions to the wave equation, inverse problems, and point configuration questions. Results on L^{p} improving estimates for the spherical averaging operator in the continuous setting have been studied classically by Littman and Strichartz in the 1970s. In contrast, it is only recently that l^{p} improving results have been obtained in the discrete setting for the spherical averaging operator. These ideas can be used to study multilinear geometric averaging operators motivated by Falconer type problems and how the underlying graph structure of the point configuration impacts the mapping properties of the operators. I conduct a study of averaging operators based on connected graphs over the integers and seeking to understand how graph structure of the underlying configuration impacts mapping properties. 

Find a preprint here.

The Erdős-Falconer distance problem, which plays an important role in arithmetic combinatorics and harmonic analysis, has been much studied in recent years. Variants of this problem include a change of setting, as in the finite field variant, which is posed by Alex Iosevich and Misha Rudnev, or a change in the object being counted, such as Passant's work on counting Erdős chains in the plane.

Palsson and Wang introduced a density variant of the Erdős-Falconer distance problem over integers via the similarity between measure and density, using the l^p improving properties of the spherical averaging operator established by Hughes and Kessler-Lacey. Following their method, I am working on a density variant of the problem of counting Erdős chains using the l^p improving results for forms based on chains.