Research

Publications and Preprints

A Maximal Extension of the Best-Known Bounds on the Furstenberg-Sárközy Theorem, Submitted (arxiv link).

Abstract: We show that if h is an integer polynomial of degree k≥2 whose image contains a multiple of q for every natural number q, known as an intersective polynomial, then any subset of {1,2,…,N} with no nonzero differences of the form h(n) has density at most a constant depending on h times (log N)^{-c log log log log N}, where c=(2 log k)^{-1}. Bounds of this type were previously known only for monomials and intersective quadratics, and this is currently the best-known bound for the original Furstenberg-Sárközy Theorem, i.e. h(n)=n^2. The intersective condition is necessary to force any density decay for polynomial difference-free sets, and in that sense our result is the maximal extension of this particular quantitative estimate. Further, we show that if and h are intersective polynomials, then any set lacking nonzero differences of the form g(m)+h(n) has density at most exp(-c(log N)^{μ}), where c=c(g,h)>0, μ=μ(deg(g),deg(h))>0, and μ(2,2)=1/2. We also include a brief discussion of sums of three or more polynomials in the final section.

Group Actions and a Multi-Parameter Falconer Distance Problem (with Kyle Hambrook and Alex Iosevich), Submitted (arxiv link).

Abstract: We provide a multi-parameter generalization of the best-known results on the Falconer Distance Problem. See the linked .pdf for a more detailed abstract.

Polynomials and Primes in Generalized Arithmetic Progressions (Revised Version)          (with Ernie Croot and Neil Lyall), Int. Math. Res. Not. (2015), no. 15, 6021-6043.

Abstract: We provide upper bounds on the density of a symmetric generalized arithmetic progression lacking nonzero elements of the form h(n) for natural numbers n, or  h(p) with p prime, for appropriate polynomials h with integer coefficients. The prime variant can be interpreted as a multi-dimensional, polynomial extension of Linnik’s Theorem. This version is an improvement of the published version. Most notably, the properness hypotheses have been removed from Theorems 2 and 3, and the numerology in Theorem 2 has been improved.

A Quantitative Result on Diophantine Approximation for Intersective Polynomials                  (with Neil Lyall), INTEGERS Volume 15A (2015), Proceedings of Integers 2013: The Erdös Centennial Conference.

Abstract: In this short note, we closely follow the approach of Green and Tao to extend the best known bound for recurrence modulo 1 from squares to the largest possible class of polynomials. The paper concludes with a brief discussion of  a consequence of this result for polynomial structures in sumsets and limitations of the method.

A Purely Combinatorial Approach to Simultaneous Polynomial Recurrence Modulo 1
(with Ernie Croot and Neil Lyall), Proceedings of the AMS 143 (2015), no. 8, 3231-3238.

Abstract:  Using purely combinatorial means we obtain results on simultaneous Diophantine approximation modulo 1 for systems of polynomials with real coefficients and no constant term.

Sárközy’s Theorem for P-Intersective Polynomials,                                                          Acta Arithmetica 157 (2013), no. 1, 69-89.

Abstract: We define a necessary and sufficient condition on a polynomial h with integer coefficients to guarantee that every set of natural numbers of positive upper density contains a nonzero difference of the form h(p) for some prime p. Moreover, we establish a quantitative estimate on the size of the largest subset of {1,2,…,N} which lacks the desired arithmetic structure, showing that if deg(h)=k, then the density of such a set is at most a constant times (log N)^{-c} for any c<1/(2k-2). We also discuss how an improved version of this result for k=2 and a relative version in the primes can be obtained with some additional known methods.

Improved Bounds on Sárközy’s Theorem for Quadratic Polynomials                                (with Mariah Hamel and Neil Lyall), Int. Math. Res. Not. (2013), no.8, 1761-1782.

Abstract: We extend the best known upper bound on the largest subset of {1,2,…,N} with no square differences to the largest possible class of quadratic polynomials.

Polynomial Differences in the Primes  (with Neil Lyall), Combinatorial and Analytic Number Theory 2011-2012, Springer Proceedings of Mathematics and Statistics vol. 101 (2014), 129-146

Abstract: We establish, utilizing the Hardy-Littlewood Circle Method, an asymptotic formula for the number of pairs of primes whose differences lie in the image of a fixed polynomial. We also include a generalization of this result where differences are replaced with any integer linear combination of two primes.

Improvements and Extensions of Two Theorems of Sárközy (Ph. D. Thesis)

Abstract: We explore quantitative improvements and extensions of two theorems of Sárközy, the qualitative versions of which state that any set of natural numbers of positive upper density necessarily contains two distinct elements which differ by a perfect square, as well as two elements which differ by one less than a prime number.

Computational Projects

Numerical Data on Questions of Erdös and Lovász (with Drewrey Lupton), in preparation

Abstract: In this note, we provide the size of the largest subset of {1,2, … ,N} with no nonzero perfect square differences for 1≤N≤243, as well as several examples of such maximally large subsets. Further, we provide the same data with `”perfect square” replaced by “one less than a prime number” for 1≤N≤400. This data relates to questions originally posed by Lovász and Erdös, respectively. In addition, we provide a detailed pseudocode explanation of the algorithms used, which we developed by adapting the methods of Gasarch, Glenn, and Kruskal on the analogous question for three-term arithmetic progressions. We also include a link to our actual Python code, which can accommodate any set of avoided differences.

The Python code for this project, including extensive comments and examples for execution, can be found in these two .txt files:

DifferencesPrep

DifferencesMain

On Squares, Primes, and More in Two-Dimensional Generalized Arithmetic Progressions (with Tim Tribone), in preparation

Abstract: First, we follow previous work of the first author with Lyall and Croot in providing a completely self-contained proof that given a natural number N, every symmetric two-dimensional generalized arithmetic progression (GAP) in [-N,N]:={-N,…,0,1,…N} that contains no nonzero perfect squares has size at most C(ε)N^{10/11+ε} for any ε>0. With this result as motivation, we provide an algorithm that, given a threshold N and a set M in [-N,N], determines for each 1 ≤ n ≤ N the largest two-dimensional symmetric GAP in [-n,n] containing no nonzero elements of M. We highlight collected data for N=100000 in the cases where M is the set of squares as well as M={p-1: p prime}. We also provide a link to our Python code, which includes comments and instructions.

The Python code for this project, including extensive comments and examples for execution, can be found in this .txt file:

2dGAP

 

UGA VIGRE Research Group on Elliptic Curves with Complex Multiplication

Torsion Points on Elliptic Curves with Complex Multiplication (with Pete Clark, Brian Cook, and James Stankewicz), International Journal of Number Theory 9 (2013), 447-479.

Abstract: We present seven theorems on the structure of prime order torsion points on CM elliptic curves defined over number fields. The first three results refine bounds of Silverberg and Prasad-Yogananda by taking into account the class number of the CM order and the splitting of the prime in the CM field. In many cases we can show that our refined bounds are optimal or asymptotically optimal. We also derive asymptotic upper and lower bounds on the least degree of a CM-point on X_1(N). Upon comparison to bounds for the least degree for which there exist infinitely many rational points on X_1(N), we deduce that, for sufficiently large N, X_1(N) will have a rational CM point of degree smaller than the degrees of at least all but finitely many non-CM points.

Computation on Elliptic Curves with Complex Multiplication (with Pete Clark, Patrick Corn, and James Stankewicz), LMS J. Comput. Math. 17 (2014), no.1, 509-535.

Abstract:  We give the complete list of possible torsion subgroups of elliptic
curves with complex multiplication over number fields of degree 1-13. Addi-
tionally we describe the algorithm used to compute these torsion subgroups
and its implementation.

Expository Articles/Notes

Waring’s Problem

Abstract: These are notes compiled from an independent study with two University of Rochester undergraduate students during the Spring 2016 semester, and a treatment of the same material in MTH 230: Number Theory during Fall 2016, in which we give a mostly self-contained exposition of the fact that if s>2^k, then every sufficiently large natural number can be written as a sum of s perfect k-th powers in close to the expected number of ways. As the notes were prepared for a course, some of the details are left as exercises, or alluded to as previously completed exercises. The notes are aimed at an undergraduate audience, and the only strict prerequisites are differential and integral calculus, a small amount of elementary number theory, and a general awareness of complex numbers. While some experience and comfort with complex exponentials and Big-O notation will help, we make an effort to introduce and discuss all required tools. (Still in progress.)

Two Theorems of Sárközy  (with Neil Lyall)

Abstract: In this note, we provide parallel expositions of two theorems of Sárközy, the qualitative versions of which state that any set of natural numbers of positive upper density necessarily contains two distinct elements which differ by a perfect square, as well as two elements which differ by one less than a prime number. We use simplified versions of Sárközy’s original methods, and the proofs are self-contained with the exception of the Weyl Inequality, minor arc estimates of Vinogradov, and the Siegel-Walfisz Theorem on primes in arithmetic progressions.

Goldbach’s Problems

Abstract: These are notes I prepared as an early career graduate student from the UGA Analysis and Arithmetic Combinatorics Learning Seminar in Fall 2009, which I organized with John Doyle and Neil Lyall. Here we show, black boxing some major components, that every sufficiently large odd integer is the sum of three primes, and almost every even integer is the sum of two primes.

Density and Substance  (undergrad friendly)

Abstract: We investigate different notions of the size of a subset of the natural numbers. This note was prepared as a final project for Pete Clark’s elementary number theory course in Spring 2007, my third year of undergraduate study at the University of Georgia, and is intended for an undergraduate audience.