Research Interests

My research involves the interplay of combinatorics, geometry, number theory, and harmonic analysis. I am largely interested in problems that say in one form or another that a sufficiently large set has sufficiently nice additive or geometric structure.

I study problems such as the finite field distance problem, finite point configurations, and sum-product phenomena. I am also interested in certain aspects of algebraic coding theory such as Steiner and Kirkman systems.

Books

"The Finite Field Distance Problem," was published in the Carus Mathematical Monographs in 2021 though the AMS Book Program. This book explores the vast literature concerning the so-called Erdös-Falconer Distance Problem in vector spaces over finite fields along with many generalizations of the problem and some related problems. Samples and more are available here.

Publications

11. The generalized k-resultant modulus set problem in finite fields (with D. Koh and Y. Pi)

- (2019) Journal of Fourier Analysis and Applications, MR3953496.

10. Cayley Digraphs Associated to Arithmetic Groups (with Y. Demiroğlu and J. Pakianathan)

- (2019) Graphs and Combinatorics, MR391496 .

9. Pairs of dot products in finite fields and rings (with S. Senger)

- (2017) Combinatorial and Additive Number Theory II, MR3754899.

8. The k-resultant modulus set problem on algebraic varieties over finite fields (with D. Koh and Y. Pi)

- (2017) Finite Fields and Their Applications, MR3705735.

7. Long paths in the distance graph over large subsets of vector spaces over finite fields (with M. Bennett, J. Chapman, D. Hart, A. Iosevich, and J. Pakianathan)

- (2016) J. Korean Math. Soc., MR3450941.

6. On the sums of any k points in finite fields (with D. Koh and Y. Pi)

- (2016) SIAM J. Discrete Math., MR3466193.

5. Results on the Erdos-Falconer Distance problem in Z_q^d for odd q

- (2015) J. Math. Anal. Appl., MR3314855.

4. Geometric configurations in the ring of integers modulo p^l (with A. Iosevich and J. Pakianathan)

- (2012) Indiana University Mathematics Journal, MR3119606.

3. A Furtsenberg-Katznelson-Weiss type-theorem on (d+1)-point configurations in sets of positive density in finite field geometries (with D. Hart, A. Iosevich, S. Senger, and I. Uriarte-Tuero)

- (2011) Discrete Math., MR2799893.

2. Sums and Products with Smooth Numbers (with William D. Banks)

- (2011) Journal of Number Theory, MR2772482.

1. Generalized Incidence Theorems, homogeneous forms, and sum-product estimates in finite fields (with D. Hart, A. Iosevich, D. Koh, and M. Rudnev)

- (2010) European J. Combin., MR2552610.

Former Ph.D. Students:

Doctoral Committee Member for Nick Elhman (2025)
Doctoral Committee Member for Eliot Courtois (2023)
Doctoral Committee Member for Adam Wagenknect (2023)
Ph.D. co-advisor to Bill Brummond (2019)
Ph.D. co-advisor to Yesim Demiroglu-Karabulat (2018 - University of Rochester)

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