A brief overview
My research interests focus on various problems within number theory and ergodic theory. 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.
One of the biggest projects I have worked on has been a long-running study of continued fractions in higher dimensions, and their connections to hyperbolic spaces. This project has been undertaken with Anton Lukyanenko.
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
- A hot spot proof of the generalized Wall's theorem (with V. Bergelson) to appear in American Mathematical Monthly.
- Towards a sharp converse of Wall's Theorem on arithmetic progressions to appear in Pacific Journal of Mathematics.
- Differencing Methods for Korobov-type exponential sums to appear in Journal d'Analyse Mathematique.
- On the binary digits of √2 #A30 Integers 18. (arXiv)
- Uncanny subsequence selections that generate normal numbers Uniform Distribution Theory 12 (2017), no. 2, 65-75. (arXiv)
- Non-trivial matrix actions preserve normality for continued fractions Compositio Mathematica 153 (2017), no. 2, 274-293. (arXiv)
- Absolutely abnormal, continued fraction normal numbers Bulletin of the Australian Mathematical Society 94 (2016), no. 2, 217-223. (arXiv)
- Normal number constructions for Cantor series with slowly growing bases (with D. Airey and B. Mance) Czechoslovak Mathematical Journal 66(141) (2016), no. 2, 465-480. (arXiv)
- New normality constructions for continued fraction expansions Journal of Number Theory 166 (2016), 424-451. (arXiv)
- Continued fraction normality is not preserved along arithmetic progressions (with B. Heersink) Archiv der Mathematik 106 (2016), no. 4, 363-370 (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 Bulletin of the London Mathematical Society 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 Arithmetica 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 Quaterly Journal of Mathematics 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 Arithmetica 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
- A proof of the infinitude of primes via continued fractions submitted
- Calculations of the invariant measure for Hurwitz continued fractions (with G. Hiary) submitted
- Ergodicity of Iwasawa continued fractions via markable hyperbolic geodesics (with A. Lukyanenko) 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
In preparation or under revision
- Squares in Polynomial Product Sequences (with Paul Spiegelhalter)