Mathematics Research

Research Articles

• On the Degree-2 Siegel Theta Series of Extremal Even Unimodular Lattices of Ranks 48, 72, 96, and 120

  Hokkaido Mathematical Journal, forthcoming
• For Which Real Quadratic Fields Is Kim's Octonary Form Universal?

  manuscripta mathematica, forthcoming
• Extremal Type II Lattices of Rank 72 are Generated by their Second Shell

  Tokyo Journal of Mathematics, forthcoming
• A Flanking Pattern in a Sum-of-Divisors Congruence

  Journal of Integer Sequences, forthcoming
• Uniform Bounds for Digit-Appending Fibonacci Walks

  The Fibonacci Quarterly, forthcoming (preprint)
• Simultaneous Niven Numbers in Arithmetic Progressions for Power-Related Bases

  Bulletin of the Australian Mathematical Society, forthcoming (preprint)
• On Digit-Appending Walks Along Linear Recurrences [with A. Mosahab, S. Boyyakar, G. Bruda, and S. J. Miller]

  Working paper, 2026
• A Fixed-Prime Criterion for Reciprocals in Missing-Digit Sets

  Working paper, 2026
• A Convexity Proof of Pohst's Inequality

  Working paper, 2026
• A Fibonacci Analogue of the Fifteen Theorem

  Working paper, 2026
• A Fourier Consequence for Lattice Kissing Numbers and Average Contacts

  Working paper, 2026
• A Harmonic Law for Reciprocal Arithmetic Fractals

  Working paper, 2026
• Saturation and No-Go Theorems for Scalar Poisson Certificates of Gaussian Mass Maximality

  Working paper, 2026
• A Proper Euler Magic Matrix of Order 5

  Working paper, 2026
• A Sharp Reverse Minkowski Inequality for the Gaussian Mass of Integral Unimodular Lattices Through Rank 32

  Working paper, 2026
• Center-Zero Magic Squares of Squares over Number Fields

  Working paper, 2026
• Consecutive Divisors of Sums of Fibonacci Polynomials: a Luca–Marques Analogue in Finite-Field Polynomial Rings

  Working paper, 2026
• Effective Lower Bounds for the Multiplicative Order of φ(n) Modulo a Lehmer Counterexample

  Working paper, 2026
• Equality in a Reverse Minkowski Shell Bound for Integral Lattices via Spherical Designs

  Working paper, 2026
• Exact Ambient Dimensions for Saturated Icubes

  Working paper, 2026
• Finite-Family Avoidance and Isolation Gaps for Quadratic Forms

  Working paper, 2026
• Long Intervals Without Distinct Multiples of the First n Positive Integers

  Working paper, 2026
• Majorization and Gaussian-Mass Maximality for Construction-A Lattices from Binary Self-Dual Codes

  Working paper, 2026
• No Nontrivial Pentagonal Cubes: The Even Case

  Working paper, 2026
• On the Number of Solutions of kσ(k) = n

  Working paper, 2026
• Partial Products, Screening, and a Matching Refinement of Pohst's Inequality

  Working paper, 2026
• Prime Values and Primitive Restrictions of Positive-Definite Hermitian Forms over Imaginary Quadratic Fields

  Working paper, 2026
• Representation of Primes in Arithmetic Progressions by Positive-Definite Integral Quadratic Forms

  Working paper, 2026
• Simultaneous Representation of Primes by Multiple Positive-Definite Integral Quadratic Forms

  In preparation, 2026
• Small Algebraic Integers and Frobenius Prime Sums

  Working paper, 2026
• Sparsity Obstructions for Abelian Power Compositions of Odd Base Degree

  Working paper, 2026
• Squared Edge Lengths of Regular Simplices with Rational Vertices

  Working paper, 2026
• Sums of Four Squares with Sandwich Constraints

  Working paper, 2026
• Timid Signed Quasifibonacci Products

  Working paper, 2026
• Uniform Finiteness for the Goormaghtigh Equation in Function Fields

  Working paper, 2026
• The Asymptotic Number of Distinct Reciprocal Subset Sums [with J. Neu]

  Working paper, 2026 (formalization and computational certificates)
• A Lattice Family with Kissing Constant τ(ℒ_n) ≥ e^2√n [with T. Laarhoven]

  Working paper, 2026 (earlier Laarhoven preprint; earlier Kominers preprint)
• Oh's 8-Universality Criterion Is Unique

  Kyungpook Mathematical Journal, 2021 (preprint)
• Configurations of Extremal Type II Codes via Harmonic Weight Enumerators [with N. D. Elkies]

  Journal de Theorie des Nombres de Bordeaux, 2019 (preprint)
• Asymptotic Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions [with D. M. Kane]

  Canadian Mathematical Bulletin, 2014 (preprint)
• Minimal S-Universality Criteria May Vary in Size [with N. D. Elkies and D. M. Kane]

  Journal de Theorie des Nombres de Bordeaux, 2013 (preprint)
• Weighted Generating Functions for Type II Lattices and Codes [with N. D. Elkies]

  Quadratic and Higher Degree Forms, 2013 (preprint)
• Refined Configuration Results for Extremal Type II Lattices of Ranks 40 and 80 [with N. D. Elkies]

  Proceedings of the American Mathematical Society, 2010 (preprint)
• On the Classification of Type II Codes of Length 24 [with N. D. Elkies]

  SIAM Journal on Discrete Mathematics, 2010 (preprint)
• On Congruence Conditions for Primality [with S. Gong]

  INTEGERS: The Electronic Journal of Combinatorial Number Theory, 2010
• Further Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions [with S. Hong]

  Proceedings of the American Mathematical Society, 2010 (preprint)
• Configurations of Extremal Even Unimodular Lattices

  International Journal of Number Theory, 2009 (preprint)
• On Universal Binary Hermitian Forms

  INTEGERS: The Electronic Journal of Combinatorial Number Theory, 2009 (preprint)
• Uniqueness of the 2-universality Criterion

  Note di Matematica, 2008
• Configurations of Rank-40r Extremal Even Unimodular Lattices (r=1,2,3) [with Z. Abel]

  Journal de Theorie des Nombres de Bordeaux, 2008 (abstract; preprint)

Surveys & Reviews

• Ozeki's Configuration Theory for Extremal Even Unimodular Lattices

  In preparation, 2026

Research Articles

• An Improved Bound for Sumsets of Thick Compact Sets via the Shapley–Folkman Theorem

  Mathematika, forthcoming
• Breaking the Fourth Wall: The Planar Centroid Banach–Mazur Diameter Is Strictly Less Than 4

  Results in Mathematics, conditionally accepted
• A Covariance Identity for Marginal Rényi Entropy of Planar Convex Bodies

  Working paper, 2026
• A Reduced Planar Body with Area Greater Than πΔ²/4

  Working paper, 2026
• Cayley Tilings, Piecewise Affine Selections, and an Exact Volume Law for the Shapley–Folkman Lemma

  Working paper, 2026
• How Thick Is the Sierpiński Triangle?

  Working paper, 2026
• Multi-Coverings by Lattice Translates of Convex Bodies

  Working paper, 2026

Research Articles

• Reed-Solomon Codes Beyond Johnson: Efficient Decoding and Smaller Cryptographic Proofs [with Q. Dao and J. Thaler]

  Working paper, 2026
• Counterexamples to Beyond-Johnson Proximity Gaps over Binary Fields [with Q. Dao, J. Thaler, and K. Z. Zheng]

  Working paper, 2026
• Improved Soundness for the Line–versus–Point Test [with J. Thaler and K. Z. Zheng]

  Working paper, 2026
• The One-and-a-Half Johnson Bound Is Tight for Proximity Gaps of General Linear Codes [with J. Thaler and K. Z. Zheng]

  Working paper, 2026

Research Articles

• On the Chromatic and Clique Numbers of the Integer Divisor Connectivity Graph

  Working paper, 2026
• Prisoners, Rooms, and Light Switches [with D. M. Kane]

  Electronic Journal of Combinatorics, 2021 (preprint)
• A Constant Bound for the Periods of Parallel Chip-firing Games with Many Chips [with P. M. Kominers]

  Archiv der Mathematik, 2010 (preprint)

Research Articles

• A (Slightly) Stronger Lower Bound on the Number of Distinct Distances from a Vertex of a Convex Polygon

  Working paper, 2026
• An Eigenvalue Proof of Hegedüs's Bound for Codes with a Single Hamming Distance

  Working paper, 2026
• Lines in the Prime Number Graph [with R. Mrazović, C. Pomerance, and P. Solé]

  Working paper, 2026
• A Universal In-Place Reconfiguration Algorithm for Sliding Cube-Shaped Robots in a Quadratic Number of Moves [with Z. Abel, H. A. Akitaya, M. Korman, and F. Stock]

  Proceedings of the 40th International Symposium on Computational Geometry (SoCG), 2024
• Hinged Dissections Exist [with T. G. Abbott, Z. Abel, D. Charlton, E. D. Demaine, and M. L. Demaine]

  Discrete & Computational Geometry, 2012 (conference version, Proceedings of the 24th Annual Symposium on Computational Geometry, 2008; preprint)
• Every Large Point Set contains Many Collinear Points or an Empty Pentagon [with Z. Abel, B. Ballinger, P. Bose, S. Collette, V. Dujmovic, F. Hurtado, S. Langerman, A. Por, and D. R. Wood]

  Graphs and Combinatorics, 2011 (extended abstract, Proceedings of the 21st Canadian Conference on Computational Geometry; preprint)
• Shape Replication Through Self-Assembly and RNase Enzymes [with Z. Abel, N. Benbernou, M. Damian, E. D. Demaine, R. Flatland, R. Schweller, and M. L. Demaine]

  Proceedings of the 2010 ACM-SIAM Symposium on Discrete Algorithms, 2010

Research Articles

• A 9/16 Exponent for the Ball–Nayar–Tkocz Reverse Entropy Power Inequality

  Working paper, 2026 (verification script)
• Phase-Sensitive Refinements of the Boundary Schwarz Lemma

  Working paper, 2026

Surveys & Reviews

• Mathematical Minutiae: i Has This Funny Property [with Z. Abel]

  The Harvard College Mathematics Review, 2008

Research Articles

• Fractal Flipbooks [with A. J. Hawksley]

  Proceedings of Bridges 2016: Mathematics, Music, Art, Architecture, Education, Culture, 2016
• Flipbook Polyhedra [with A. J. Hawksley]

  Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture, 2013
• Irrational Roots Revisited

  The Mathematical Gazette, 2010
• Improved Bounds on the Sizes of S.P Numbers [with P. M. Kominers]

  The Mathematical Gazette, 2010
• A Categorical Construction of Ultrafilters [with D. Litt and Z. Abel]

  Rocky Mountain Journal of Mathematics, 2010 (preprint)

Surveys & Reviews

• 10/24/26: Factoring a Pythagorean Date

  Working paper, 2026
• Palindromes on the τ-circle: A Note for Palindrome Tau Day, 6/28/26

  Working paper, 2026
• Finding Matrices that Satisfy Functional Equations

  College Mathematics Journal, 2009 (preprint; abstract)

Public Writing

• Why Today Is Math Geeks' Favorite Friday the 13th in a Century

  Every, 2021
• Math Geeks Must Wait 100 Years for a Day Like This

  Bloomberg Opinion, 2018