Research


Most of my research lies in the fields of analytic and combinatorial number theory with tools from analysis such as Fourier analysis and ergodic theory. However, I view myself as a problem solver and strive to learn new mathematics and work beyond the intersection of these fields. I have worked in other areas of combinatorics and analytic number theory such as Diophantine approximation, equidistribution theory, probabilistic combinatorics, and applications to computer science. I am particularly interested in the function field setting, in which things are usually nicer and one has extra tools (e.g., GRH, linear algebra, the polynomial method) but extra difficulty also arises (e.g. caused by the totally disconnected topology or the finite characteristic).

My research is currently supported by grant DMS-1702296 from the NSF. In 2017-2018, I was also supported by a Ralph E. Powe Junior Faculty Enhancement Award from Oak Ridge Associated Universities.

The following are my publications and preprints by topics (the local version may differ slightly from the printed version). (Note: if the math does not parse, click here.)


Patterns in the primes [introduction]
[24] Multidimensional configurations in the primes with shifted prime steps
with A. Lępreprint.
[summary] [pdf] [journal]
[10] Polynomial configurations in the primes
with J. Wolf, Int. Math. Res. Not. (2014) 2014 (23): 6448--6473.
[summary] [pdf] [journal]
[7] Partition regularity and the primes
C. R. Math. Acad. Sci. Paris 350 (2012), no. 9--10, 439--441.
[summary] [pdf] [journal]
[2] Green-Tao theorem in function fields
Acta Arith. 147 (2011), 129--152.
[summary] [pdf] [journal]
Arithmetic and combinatorial structures in difference sets [introduction]
[22] On theorems of Wirsing and Sanders
with Z. GeActa Arithmetica 189 (2019), no. 4, 381--390.
[summary] [pdf] [journal]
[20] A bilinear Bogolyubov theorem
with P-Y. BienvenuEuropean Journal of Combinatorics 77, March 2019, 102--113.
[summary] [pdf] [journal]
[6] Problems and results on intersective sets
Combinatorial and Additive Number Theory: CANT 2011 and 2012, Springer Proceedings in Mathematics & Statistics (2014).
[summary] [pdf] [book]
[5] On sets of polynomials whose difference set contains no squares
with Y-R. Liu, Acta Arith. 161 (2013), 127--143.
[summary] [pdf] [journal]
[4] Difference sets and the irreducibles in function fields
with C.V Spencer, Bull. Lond. Math. Soc. 43 (2011), no. 2, 347--358.
[summary] [pdf] [journal]
[3] Intersective polynomials and the primes
Journal of Number Theory 130 (2010), Issue 8, 1705--1717.
[summary] [pdf] [journal]
Diophantine approximation [introduction]
[13] Multiplicatively badly approximable matrices in fields of power series
with J. D. Vaaler, Proc. Amer. Math. Soc. 143 (2015), no. 9, 3791--3800.
[summary] [pdf] [journal]
[12] Intersective polynomials and Diophantine approximation, II
with C. V. Spencer, Monatsh. Math. 177 (2015), no. 1, 79--99.
[summary] [pdf] [journal]
[11] Sums of products of fractional parts
with J. D. Vaaler, Proc. London Math. Soc. (3) 111 (2015), no. 3, 561--590.
[summary] [pdf] [journal]
[8] Intersective polynomials and Diophantine approximation
with C. V. Spencer, Int. Math. Res. Not. (2014) 2014 (5): 1153--1173.
[summary] [pdf] [journal]
Exponential and character sum estimates [introduction]
[21] Linear and quadratic uniformity of the Möbius function over $\mathbf{F}_q[t]$
with P-Y. BienvenuMathematika 65 (2019), Issue 3, 505--529.
[summary] [pdf] [journal]
[19] A note on a chacracter sum in finite fields
with A. Bhowmick and Y-R. Liu, Finite Fields and Their Applications 46 (2017), 247--254.
[summary] [pdf] [journal]
Equidistribution [introduction]
[18] Van der Corput sets with respect to compact groups
with M. Kelly, Arch. Math. (Basel) 110 (2018), no. 4, 343--349.
[summary] [pdf] [journal]
[14] Equidistribution of polynomial sequences in function fields, with applications
with Y-R. Liu, preprint.
[summary] [pdf] [journal]
[9] Uniform dilations in higher dimensions
with M. Kelly, J. London Math. Soc. (2013) 88 (3): 925--940.
[summary] [pdf] [journal]
Miscellaneous number theory, combinatorics
and applications to computer science [introduction]
[25] On additive bases in infinite abelian semigroups
with P.-Y. Bienvenu and B. Girard, preprint.
[summary] [pdf] [journal]
[23] Essential components in vector spaces in finite fields
with Z. Ge, preprint.
[summary] [pdf] [journal]
[17] Additive bases in groups
with V. Lambert and A. Plagne, Israel J. Math. 217 (2017), no. 1, 383--411.
[summary] [pdf] [journal]
[16] On primitive elements in finite fields of low characteristic
with A. Bhowmick, Finite Fields and Their Applications 35 (2015) 64--77.
[summary] [pdf] [journal ]
[15] Deterministic extractors for additive sources
with A. Bhowmick, A. Gabizon and D. Zuckerman, ITCS (Innovations in Theoretical Computer Science) 2015.
[summary] [pdf] [conference]
[1] On a question of Sárközy on gaps of product sequences
with J. Cilleruelo, Israel J. Math. 179 (2010), 285--295.
[summary] [pdf] [journal]