A brief overview
My research interests focus on various problems within number theory, with an emphasis on analytic, elementary/combinatorial, and ergodic methods. My work frequently features asymptotic analysis, continued fractions, and exponential sums. My advisors were Florin Boca and Kevin Ford.
I have done several studies into continued fractions, investigating how sometimes small (and sometimes big) changes in the usual Regular Continued Fractions alter the various properties one expects a continued fraction expansion to have. Valentina Masarotto's Master Thesis is a useful and well-written introduction to this topic, and Alexey Ustinov has written several great papers in this area.
My high school mathematics teacher, Dr. John Gorman, helped develop my love of normal numbers, which I have continued to study to this day. Enthusiasts of this topic may find Bill Mance's work on normal numbers quite intriguing.
Papers
Published
- Non-trivial matrix actions preserve normality for continued fractions accepted at Compositio Mathematica.
- New normality constructions for continued fraction expansions accepted at the Journal of Number Theory
- Absolutely abnormal, continued fraction normal numbers accepted at the Bulletin of the Australian Mathematical Society
- Continued fraction normality is not preserved along arithmetic progressions (with B. Heersink) published at Archiv der Mathematik, currently online only (arXiv)
- Diophantine properties of continued fractions on the Heisenberg group International Journal of Number Theory 12 (2016), no. 2, 541-560 (arXiv)
- Normality preserving operations for Cantor series expansions and associated fractals part II (with D. Airey and B. Mance) New York Journal of Mathematics 21 (2015), 1311-1326 (arXiv)
- Besicovitch, Bisection, and the normality of 0.(1)(4)(9)(16)(25)… (with P. Pollack) The American Mathematical Monthly 122 (2015), no. 8, 757-765 (arXiv)
- Lagrange's Theorem for continued fractions on the Heisenberg group Bull. London Math. Soc. 47 (2015), no. 5, 866-882 (arXiv)
- Some normal numbers generated by arithmetic functions (with P. Pollack) Canadian Mathematical Bulletin 58 (2015), no. 1, 160-173. (arXiv)
- Continued fractions on the Heisenberg group (with A. Lukyanenko) Acta Arith. 167 (2015), 19-42 (arXiv)
- A Simpler Normal Number Construction for Simple Lüroth Series Article 14.6.1 The Journal of Integer Sequences Vol. 17 (2014)
- Error term improvements for van der Corput transforms Q. J. Math 65 (2014), no. 4, 1461-1502.
- On an incomplete argument of Erdos on the irrationality of Lambert series #A58 Integers 13 (2013)
- The normality of digits in almost constant additive functions Monatshefte für Mathematik 171 no. 3-4, pp. 481-497 (2013). (arXiv)
- On certain statistical properties of continued fractions with even and with odd partial quotients (with Florin P. Boca) Acta Arithmetica 156 no. 3, pp. 201-221 (2013). (arXiv)
- On non-intersecting arithmetic progressions (with Regis de la Breteche and Kevin Ford) Acta Arith. 157, pp. 381-392 (2013).
- On Multiplicative Functions with Bounded Partial Sums #A19 Integers 12 (2012). (arXiv)
- Containment in (s,t)-core partitions Undergraduate thesis for the University of Oregon Clark Honors College, under the direction of Marie Vitulli. (2008)
Forthcoming
- Uncanny subsequence selections that generate normal numbers submitted
- Differencing Methods for Korobov-type exponential sums submitted
- Intrinsic Diophantine approximation in Carnot groups and in the Siegel model of the Heisenberg group (with A. Lukyanenko) submitted
- On the joint normality of certain digit expansions submitted
- Normal number constructions for Cantor series with slowly growing bases (with D. Airey and B. Mance) submitted
In preparation or under revision
- Squares in Polynomial Product Sequences (with Paul Spiegelhalter)