Research

My research focus lies in number theory. While I am interested in various problems, the primary part of my work is concerned with the study of the Mahler measure on the multiplicative group of non-zero algebraic numbers. Among other things, I look to exploit the Mahler measure's relationship to the Weil height in order to better understand its behavior and the behavior of its generalizations.  My Ph.D. dissertation and related articles identified various lower bounds on the Mahler measure by expressing the Weil height in terms of auxiliary polynomials.

Since that time, I have worked extensively with certain metric versions of the Mahler measure originally developed by Arturas Dubickas and Chris Smyth in 2001.  My research often explores relationships between the metric Mahler measures and other areas of mathematics including linear programming, graph theory, real analysis, functional analysis, Diophantine approximation, and algebraic number theory.  My most recent articles construct various dual representation theorems arising from the work of Daniel Allcock and Jeffrey Vaaler.  For further information, please see my full list of articles below.

23. The correspondence between consistent maps and measures on the places of Q-bar 

• Preprint (2025).

22. A classification of Q-linear maps from the group of non-zero algebraic numbers modulo torsion to R

• Preprint (2025).

21. Consistent maps and their associated dual representation theorems 

• Colloq. Math. 178 (2025), no. 1, 11–30.  https://doi.org/10.4064/cm9180-3-2025

20. A classification of  Q-valued linear functionals on the group of non-zero algebraic numbers modulo units 

• Acta Arith. 205 (2022), no.4, 341-370.  https://doi.org/10.4064/aa211123-30-8

19. A polynomial time test to detect numbers with many exceptional points (with R. Carpenter)

1. • • Int. J. Number Theory 17 (2021), no. 4, 973-990.  https://doi.org/10.1142/S1793042121500184

18. Direct limits of adele rings and their completions (with J.P. Kelly)

• 1. • Rocky Mountain J. Math. 50 (2020), no. 3, 1021-1043.  https://doi.org/10.1216/rmj.2020.50.1021

17. Metric Mahler measures over number fields

1. • • Acta Math. Hungar. 154 (2018), no. 1, 105-123.  https://doi.org/10.1007/s10474-017-0770-y

16. Counting exceptional points for rational numbers associated to the Fibonacci sequence

1. • • Period. Math. Hungar. 75 (2017), no. 2, 221-243.  https://doi.org/10.1007/s10998-017-0189-9

15. Continued fraction expansions in connection with the metric Mahler measure

1. • • Monatsh. Math. 181 (2016), no. 4, 907-935.  https://doi.org/10.1007/s00605-016-0900-6

14. Optimal factorizations of rational numbers using factorization trees (with T.J. Strunk)

• 1. • Int. J. Number Theory 11 (2015), no. 3, 739-769.  https://doi.org/10.1142/S1793042115500402

13. Metric heights on an Abelian group

1. • • Rocky Mountain J. Math. 44 (2014), no. 6, 2075-2091.  https://doi.org/10.1216/RMJ-2014-44-6-2075

12. Two inequalities on the areal Mahler measure (with K.K. Choi)

• • • Illinois J. Math. 56 (2012), no. 3, 525-534.  https://projecteuclid.org/euclid.ijm/1391178550

11. The t-metric Mahler measures of surds and rational numbers (with J. Jankauskas)

• 1. • Acta Math. Hungar. 134 (2012), no. 4, 481-498.  https://doi.org/10.1007/s10474-011-0126-y

10. Polynomials whose reducibility is related to the Goldbach conjecture (with. P. Borwein, K.K. Choi and G. Martin)

• 1. •  JP J. Algebra Number Theory Appl. 26 (2012), no. 1, 33-63.

9. The parametrized family of metric Mahler measures

• 1. • J. Number Theory 131 (2011), no. 6, 1070-1088.  https://doi.org/10.1016/j.jnt.2011.01.003

8. A collection of metric Mahler measures

• • • J. Ramanujan Math. Soc. 25 (2010), no. 4, 433-456.

7. The finiteness of computing the ultrametric Mahler measure

• 1. • Int. J. Number Theory 6 (2010), no. 8, 1731-1753.  https://doi.org/10.1142/S1793042110003745

6. The infimum in the metric Mahler measure

• • • Canad. Math. Bull. 54 (2011), 739-747.  https://doi.org/10.4153/CMB-2011-028-x

5. On the non-Archimedean metric Mahler measure (with P. Fili)

• 1. •  J. Number Theory 129 (2009), no. 7, 1698-1708.  https://doi.org/10.1016/j.jnt.2008.12.009

4. Estimating heights using auxiliary functions

• 1. • Acta Arith. 137 (2009), no. 3, 241-251.  https://doi.org/10.4064/aa137-3-5

3. The Weil height in terms of an auxiliary polynomial

• 1. • Acta Arith. 128 (2007), no. 3, 209-221.  https://doi.org/10.4064/aa128-3-2

2. Lower bounds on the projective heights of algebraic points

• 1. • Acta Arith. 125 (2006), no. 1, 41-50.  https://doi.org/10.4064/aa125-1-4

1. Constructing almost excellent unique factorization domains (with J. Bryk, S. Mapes and G. Wang)

• 1. • Comm. Algebra 33 (2005), no. 5, 1321-1336.  https://doi.org/10.1081/AGB-200058363