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

该项目旨在研究正整数的递增序列，如偶数、奇数、素数和斐波那契数。我们将通过探索互补序列来研究递增序列，即每个整数都出现在其中一个序列中而两个序列中都不出现整数的序列。任何正整数的递增序列都可以被认为是互补对的一部分，通过取另一个集合为不在给定递增序列中的所有正整数。互补序列在其他领域也有应用。例如，所谓的Beatty序列在纯数学和应用数学、音乐、生物学、计算机图形学、线性滤波器和准晶体学中都有应用。数学家使用贪婪算法，如最小排除（MEX）算法，来生成互补序列。在之前的工作中，PI发现了使用MEX算法生成Beatty序列和连分数之间的一些令人惊讶的联系。本项目将这些发现扩展到一般互补序列。本研究的意义在于揭示了正整数的新性质，并为使用MEX算法和连分式研究的问题提供了见解。这些发现也为互补序列的应用提供了启示。此外，PI将参与提高数学多样性的教育和推广活动，包括组建一个本科研究小组，指导史密斯学院（女性数学中心所在地）攻读数学博士学位的本科学生，以及为初高中学生开展数学推广活动。PI也将继续他的工作，指导和接触在美国的多米尼加人后裔。更准确地说，本研究项目可以描述如下：通过应用MEX算法和生成互补集，获得了数论、组合学、图论、组合博弈论和理论计算机科学的经典结果和最新进展。最近，PI引入了一种新的MEX算法的推广，令人惊讶的是，它揭示了应用MEX算法生成Beatty互补序列相当于在无理数的连分数前加上数字。PI还表明，迭代MEX算法会产生一个动力系统，其轨道具有参数族的不动点，这些不动点是二次无理数。这些结果自然导致了这样一个性质：当应用MEX生成互补序列时，偶数和奇数正整数是不变的。本课题的目标是研究互补序列、MEX算法、连分式和动力系统之间的这些联系，并将其推广到一般互补序列，其中包括所有正整数的递增序列。我们将使用组合方法和标准技术，从分析和可加数论，以下策略已经被PI采用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.