Hans Parshall

My research interests are in the field of additive combinatorics, which brings together combinatorial number theory, harmonic analysis, and ergodic theory. I am particularly interested in the area of density Ramsey theory, where the typical problem concerns what "structure" must appear in all "large" sets. My work typically involves applications of Fourier analysis (and its generalizations) to problems in number theory and combinatorics.

Publications

  1. On the quotient set of the distance set (arXiv preprint)
    (with Alex Iosevich and Doowon Koh)
    • The quotient set of the distance set of every \(E \subseteq \mathbf{F}_q^2\) with \(|E| > 9q\) is equal to \(\mathbf{F}_q\).

  2. Embedding distance graphs in finite field vector spaces (arXiv preprint)
    (with Alex Iosevich)
    • Large subsets of \(\mathbf{F}_q^{2t}\) contain isometric copies of all distance graphs of maximum degree \(t\).

  3. Spherical configurations over finite fields (preprint)
    (with Neil Lyall and Ákos Magyar)
    to appear in Amer. J. Math.
    • Large subsets of \(\mathbf{F}_q^{10}\) contain isometric copies of all spherical quadrilaterals.

  4. Simplices over finite fields (arXiv preprint)
    Proc. Amer. Math. Soc. 145 (2017), 2323-2334
    • Large subsets of \(\mathbf{F}_q^{k + 1}\) contain isometric copies of every \(k\)-simplex from \(\mathbf{F}_q^k\).

  5. Small gaps between configurations of prime polynomials (arXiv preprint)
    J. Number Theory 162 (2016), 35–53
    • There are arbitrarily large affine subspaces in \(\mathbf{F}_q[t]\) consisting of twin prime polynomials.

  6. Primes represented by binary quadratic forms (PDF)
    (with Pete L. Clark, Jacob Hicks, and Katherine Thompson)
    Integers 13 (2013), A37
    • Primes represented by idoneal quadratic forms are determined by explicit congruence conditions.

I also have a few old expository notes on topics in arithmetic combinatorics.