My research focus lies in number theory, spanning roughly three distinct phases:
Functional Analysis & Number Theory (Post-Tenure; 2021-Present): My most recent articles investigate an emerging intersection of functional analysis and number theory, building directly on the work of Daniel Allcock and Jeffrey Vaaler. This track focuses heavily on L^p spaces, Riesz-type dual representation theorems, and their applications to the algebraic properties of numbers.
Metric Mahler Measures (Postdoc and Tenure-Track; 2009-2021): This direction grew out of my extensive work with certain metric versions of the Mahler measure, originally developed by Arturas Dubickas and Chris Smyth in 2001. My research in this phase explores the relationships between these metric Mahler measures and diverse areas of mathematics—including linear programming, graph theory, real analysis, functional analysis, Diophantine approximation, and algebraic number theory.
Lower Bounds on Heights (Early Career; 2005-2009): The foundation of my scholarly work, beginning with my Ph.D. dissertation and related early articles, focused on classical lower bounds on heights. This work looked to exploit the Mahler measure's relationship to the Weil height on the multiplicative group of non-zero algebraic numbers. By expressing the Weil height in terms of auxiliary polynomials, these early projects identified various lower bounds to better understand the behavior of the Mahler measure and its generalizations.
For further information on specific projects, please see my full list of publications below (* indicates an undergraduate student coauthor).
26) A Haar-like uniqueness property for measures on the places of ℚ̅, Preprint, 2026.
25) The correspondence between consistent maps and measures on the places of ℚ̅, New Zealand J. Math., to appear, 2026.
24) A classification of ℚ-linear maps from the group of non-zero algebraic numbers modulo torsion to ℝ, Canad. Math. Bull. 69 (2026), no. 2, 609-927. https://doi.org/10.4153/S0008439525101343
23) Consistent maps and their associated dual representation theorems, Colloq. Math. 178 (2025), no. 1, 11–30. https://doi.org/10.4064/cm9180-3-2025
22) The connection between consistent maps and measures (with L. Aberg*), PUMP J. Undergrad. Res. 7 (2024), 69-78. https://doi.org/10.46787/pump.v7i0.4030
21) 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
20) A polynomial time test to detect numbers with many exceptional points (with R. Carpenter), Int. J. Number Theory 17 (2021), no. 4, 973-990. https://doi.org/10.1142/S1793042121500184
19) Direct limits of adele rings and their completions (with J.P. Kelly), Rocky Mountain J. Math. 50 (2020), no. 3, 1021-1043. https://doi.org/10.1216/rmj.2020.50.1021
18) Rotation Symmetries of Sequential Matrices with Applications to the Jacobi Symbol (with Y. Ayub*), PUMP J. Undergrad. Res. 2 (2019), 87-94. https://doi.org/10.46787/pump.v2i0.500
17) Metric Mahler measures over number fields, 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, 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, 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*), Int. J. Number Theory 11 (2015), no. 3, 739-769. https://doi.org/10.1142/S1793042115500402
13) Metric heights on an Abelian group, 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), 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), JP J. Algebra Number Theory Appl. 26 (2012), Vol. 26, no. 1, 33-63.
9) The parametrized family of metric Mahler measures, 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, 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), 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, 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, 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, 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), Comm. Algebra 33 (2005), no. 5, 1321-1336. https://doi.org/10.1081/AGB-200058363