LEAPS-MPS: Number-Theoretic and Combinatorial Properties of Increasing Sequences of Positive Integers

Smith College, Northampton MA

Investigators

Abstract

This project aims to study increasing sequences of positive integers, such as the evens, the odds, prime numbers, and Fibonacci numbers. We will study increasing sequences by exploring complementary sequences, that is, sequences such that every integer appears in one of them and no integer appears in both. Any increasing sequence of positive integers can be thought of as part of a complementary pair by taking the other set to be all positive integers that are not in the given increasing sequence. Complementary sequences also may have applications in other areas. For instance, the so-called Beatty sequences have applications in pure and applied mathematics, music, biology, computer graphics, linear filters, and quasi-crystallography. Mathematicians use greedy algorithms, such as the minimum Excluded (MEX) algorithm, to generate complementary sequences. In previous work, the PI discovered some surprising connections between the use of the MEX algorithm to generate Beatty sequences and continued fractions. This project will extend these findings to general complementary sequences. The significance of this research lies in uncovering new properties of positive integers and providing insights into problems studied using the MEX algorithm and continued fractions. These discoveries could also shed light on the applications of complementary sequences. Additionally, the PI will engage in educational and outreach activities enhancing diversity in mathematics, including forming an undergraduate research group and mentoring math-PhD-bound post baccalaureate students at Smith College (home to the Center for Women in Mathematics), and conducting math outreach for middle/high-school students. The PI will also continue with his work in mentorship and engagement with people of Dominican descent in the US. More precisely, this research project can be described as follows: Classical results and recent developments in number theory, combinatorics, graph theory, combinatorial game theory and theoretical computer science are obtained by applying the MEX algorithm and generating complementary sets. Recently the PI introduced a novel generalization of the MEX algorithm which reveals, surprisingly, that applying the MEX algorithm to generate Beatty complementary sequences is equivalent to prepending digits to the continued fractions of irrational numbers. The PI has also shown that iterating the MEX algorithm gives rise to a dynamical system whose orbits have a parametric family of fixed points that are quadratic irrationals. These results lead naturally to the property that the even and the odd positive integers are invariant when applying the MEX to generate complementary sequences. The goal of this project is to study these connections between complementary sequences, the MEX algorithm, continued fractions, and dynamical systems, extending them to general complementary sequences, which includes all increasing sequences of positive integers. We will use combinatorial methods and standard techniques from analytic and additive number theory, following strategies that have been already employed by the PI. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.