These are all of the papers that I have worked on which are in the process of being published or have been published. Relevant links are included, both to journals, the arxiv, and pdfs (when appropriate).

The papers are organized with the most recent near the top and the oldest near the bottom within each section. The titles are also links to the pdfs.

Published

• Distinct Angle Problems and Variants , with Henry L. Fleischmann, Hongyi B. Hu, Steven J. Miller, Eyvindur A. Palsson, Ethan Pesikoff, and Charles Wolf. PDF . Discrete and Computational Geometry. https://doi.org/10.1007/s00454-022-00466-w . 2023.
  • Studies the Erdos distinct angle problem and variants on finding the minimum number of distinct angles between nnn points in the plane, which were previously unstudied despite the ubiquitousness of the Erdos distinct distance problem.
  • Research done as part of the 2021 SMALL REU under the advisement of Steven J. Miller, Eyvindur A. Palsson, and Charles Wolf.
• The Bergman Game , with Benjamin Baily, Justine Dell, Irfan Durmic, Henry L. Fleischmann, Isaac Mijares, Steven J. Miller, Ethan Pesikoff, Luke Reifenberg, Alicia Smith Reina, Yingzi Yang. PDF . The Fibonacci Quarterly (Proceedings of the 20th Conference), 60.5, 18-38. https://www.fq.math.ca/Papers1/60-5/baily.pdf . 2022.
  • Defines and studies the Bergman Game, which decomposes positive integers into their base-$\varphi$ expansions.
  • Research done as part of the 2021 SMALL REU under the advisement of Steven J. Miller.
• Limiting Spectral Distributions of Families of Block Matrix Ensembles , with Teresa Dunn, Henry L. Fleischmann, Simran Khunger, Steven J. Miller, Luke Reifenberg, Alexander Shashkov, and Stephen Willis. PDF . The PUMP Journal of Undergraduate Research, Volume 5, 122-147. 2022.
  • Describes a new operation on matrices $\operatorname{swirl}(A, X)$ and its effect on the limiting spectral distribution of an ensemble, as well as providing a novel combinatorial proof that the ensemble of circulant Hankel matrices have the Rayleigh distribution as a limiting spectral measure.
  • Research done as part of the 2021 SMALL REU under the advisement of Steven J. Miller.
• Unexpected Biases between Congruence Classes for Parts in $k$-indivisible Partitions , with Misheel Otgonbayar. PDF . Journal of Number Theory. 248, 310-342. https://doi.org/10.1016/j.jnt.2023.01.006 2023.
  • Using the circle method, we prove an asymptotic for $D_k^\times(r,t;n)$, which we define to be the number of parts among all $k$-indivisible partitions (those where no part is divisible by $k$) of $n$ which are congruent to $r$ mod $t$, when $k,t$ are coprime. We then observe that this implies the parts are asymptotically equidistributed; however, there is a bias in the second order term. However, unlike previous biases of this type, the bias is highly unpredictable. We show that the bias reverts to the natural bias towards lower congruence classes for $k > \frac{6(t^2-1)}{\pi^2}$, and we explore the intricate properties of this bias for $k < \frac{6(t^2-1)}{\pi^2}$.
  • Research done as part of the 2022 UVA REU under the advisement of William Craig and Ken Ono.

Preprints

• [Parts in k-indivisible Partitions Always Display Biases between Residue Classes] , with Misheel Otgonbayar. PDF . Submitted to the Journal of Number Theory. 2023
  • Inspired by our previous paper Unexpected Biases between Congruence Classes for Parts in $k$-indivisible Partitions (see above), we examined the second-order term in the asymptotic for $D_k^\times(r,t;n)$ derived in that paper in more detail. In this paper, we confirm our earlier conjecture that there are no “ties” (i.e., equalities) in this asymptotic for different congruence classes. To obtain this result, we reframe this question in terms of $L$-functions, and we then employ a nonvanishing result due to Baker, Birch, and Wirsing to conclude that there is always a bias towards one congruence class or another modulo $t$ among all parts in $k$-indivisible partitions of $n$ as $n$ becomes large.
• Biases among Congruence Classes for Parts in $k$-regular partitions , with Misheel Otgonbayar. PDF . Submitted to the Mathematical Proceedings of the Cambridge Philosophical Society. 2022.
  • Using the circle method, we prove an asymptotic for $D_k(r,t;n)$, which we define to be the number of parts among all $k$-regular partitions (those whose parts have multiplicity at most $k - 1$) of $n$ which are congruent to $r$ mod $t$. We then observe that this implies the parts are asymptotically equidistributed; however, there is a bias in the second order term towards the lower congruence classes. We make this explicit and show that for $3 \leq k \leq 10, 2 \leq t \leq 10$ that for all $n \geq 1$ we have $D_k(r,t;n) > D_k(s,t;n)$ when $r < s$.
  • Research done as part of the 2022 UVA REU under the advisement of William Craig and Ken Ono.
• Irreducibility Over the Max-Min Semiring , with Benjamin Baily, Justine Dell, Henry L. Fleischmann, Steven J. Miller, Ethan Pesikoff, and Luke Reifenberg. 2021. PDF . Submitted to the International Journal of Algebra and Computation.
  • Proves that over the base-$b$ max-min semiring that almost all polynomials of a given degree are irreducible as the degree tends to $\infty$. Furthermore this paper proves that almost all power series over the max-min semiring are asymptotically irreducible.
  • Research done as part of the 2021 SMALL REU under the advisement of Steven J. Miller.
• The Generalized Bergman Game , with Benjamin Baily, Justine Dell, Irfan Durmic, Henry L. Fleischmann, Isaac Mijares, Steven J. Miller, Ethan Pesikoff, Luke Reifenberg, Alicia Smith Reina, Yingzi Yang. 2021. PDF arxiv . Submitted to the CANT 2022 Conference Proceedings.
  • Defines and studies the Generalized Bergman Game, which decomposes positive integers into their base-$\beta$ expansions, for $\beta$ a root of a non-increasing positive linear recurrence. Proves that the game terminates in $\Theta(n^2)$ time in the worst-case, where nnn is the number of tokens in the initial state.
  • Research done as part of the 2021 SMALL REU under the advisement of Steven J. Miller.

Drafts

• Large Sets are Sumsets , with Benjamin Baily, Justine Dell, Sophia Dever, Adam Dionne, Henry L. Fleischmann, Steven J. Miller, Leo Goldmakher, Gal Gross, Ethan Pesikoff, Huy Pham, Luke Reifenberg, and Vidya Venkatesh. 2021. DRAFT .
  • Proves an explicit upper bound of $\left\lvert B\right\rvert - \alpha_d \log \left\lvert B\right\rvert$ on the size of irreducible subsets of a box $B$ in $\mathbb{N}^d$ for some computable constant $\alpha_d$. Furthermore this paper proves that this bound is asymptotically tight by providing a construction of an irreducible subset of any box $B$ in $\mathbb{N}^d$ of size $\left\lvert B\right\rvert - \beta_d \log \left\lvert B\right\rvert$ for some computable constant $\beta_d$.
  • Research done as part of the 2021 SMALL REU under the advisement of Steven J. Miller and Leo Goldmakher.
• The Complexity of the Zeckendorf Graph Game , with Benjamin Baily, Justine Dell, Henry L. Fleischmann, Steven J. Miller, Ethan Pesikoff, and Luke Reifenberg. 2021. DRAFT .
  • Defines the Zeckendorf Graph Game, as well as the Generalized Zeckendorf Graph Game, and shows that the problem of deciding who has a winning strategy from a given initial state is PSPACE-Complete.
  • Research done as part of the 2021 SMALL REU under the advisement of Steven J. Miller.