Research

My primary research interests lie in the intersection of harmonic analysis and geometric measure theory.  These fields have proved to be closely tied to each other; major results about the geometry of fractal sets have been proved using tools and techniques from harmonic analysis.  My research centers around two major sticking points between these fields: the existence of point configurations in sufficiently large fractal sets, and measure theoretic properties of projections.  I also work on related problems in geometric combinatorics.

Publications and Preprints (listed newest to oldest)

• Prescribed projections for a general class of maps and efficient covering by variable plane curves (joint with Alan Chang, Robert Fraser, Mark Meyer, and Krystal Taylor).  (arXiv)

• VC-dimension of distance graphs in finite field geometries (joint with Brian McDonald Bakas and Moustapha Diallo*).

• Favard length and generalized projections (joint with Izabella Laba and Krystal Taylor).  (arXiv)

• Extremal graph theory and point configurations in Ahlfors-David regular sets.  (arXiv)

• A non-linear Roth theorem for thick Cantor sets (joint with Micah Nguyen*).  (arXiv)

• The VC-dimension and point configurations in R^d (joint with Alex Iosevich, Akos Magyar, and Brian McDonald), accepted for publication in Journal of Mathematical Analysis and Applications. (arXiv)

• Prescribed projections and efficient coverings of sets by curves (joint with Alan Chang and Krystal Taylor), accepted for publication in in Analysis & PDE.  (arXiv)

• Point configurations in sets of sufficient topological structure and a topological Erdos similarity conjecture (joint with Krystal Taylor), Res. Math. Sci. 12 (2025), no. 3, Paper No. 45.  (arXiv | Journal)

• Infinite constant gap length trees in products of thick Cantor sets (joint with Krystal Taylor), Proc. Roy. Soc. Edinburgh Sect. A 154 (2024), no. 5, 1336–1347.  (arXiv | Journal)

• Finite point configurations in products of thick Cantor sets and a robust nonlinear Newhouse Gap Lemma (joint with Krystal Taylor), Math. Proc. Cambridge Philos. Soc. 175(2023), no. 2, 285-301.  (arXiv | Journal)

• Volumes spanned by k-point configurations in R^d (joint with Belmiro Galo), J. Geom. Anal. 32 (2022), no. 1, 23.  (arXiv | Journal)

• Areas spanned by point configurations in the plane, Proc. Amer. Math. Soc., 149(5):2035-2049, 2021.  (arXiv | Journal)

• Distinct distances from points on a circle to a generic set (joint with Brian McDonald, Jonathan Passant, and Anurag Sahay), Integers 21 (2021), Paper no. A55.  (arXiv | Journal)

• Congruence classes of large configurations in vector spaces over finite fields, Funct. Approx. Comment. Math., Volume 62, Number 1 (2020), 131-141. (arXiv | Journal)

• Areas of triangles and SL_2 actions in finite rings, BULLETIN of the L.N. Gumilyov Eurasian National University Mathematics Series, Computer science, Mechanics, No.2 (127) / 2019. (arXiv | Journal)

*Indicates undergraduate co-author

My Collaborators (# of papers)

• Krystal Taylor (6)

• Brian McDonald Bakas (3)

• Alan Chang (2)

• Moustapha Diallo* (1)

• Robert Fraser (1)

• Belmiro Galo (1)

• Alex Iosevich (1)

• Izabella Laba (1)

• Akos Magyar (1)

• Mark Meyer (1)

• Micah Nguyen* (1)

• Jonathan Passant (1)

• Anurag Sahay (1)

*Indicates undergraduate co-author