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

LEAPS-MPS：正整数递增序列的数论和组合性质

• 批准号: 2316986
• 负责人: Geremias Polanco
• 金额: $ 24.54万
• 依托单位: Smith College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2316986&HistoricalAwards=false
• 关键词: LEAPS; MPS; Number; Theoretic; Combinatorial

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.

这个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，互补序列是指每个整数都出现在其中一个序列中而没有整数出现在两个序列中的序列。任何正整数的递增序列都可以被认为是一个互补对的一部分，只要把另一个集合看作是不在给定递增序列中的所有正整数。互补序列也可以应用于其他领域。例如，所谓的贝蒂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有应用。数学家使用贪婪算法，例如最小排除（MEX）算法，来生成互补序列。在以前的工作中，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。这个项目将把这些发现扩展到一般的互补序列。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。这些发现也可以为互补序列的应用提供启发。此外，PI将参与教育和推广活动，提高数学的多样性，包括形成一个本科研究小组和指导史密斯学院（数学女性中心的所在地）的数学博士学位后的学生，并为初中/高中学生进行数学推广。PI还将继续与美国的多米尼加裔人进行指导和接触。更准确地说，这个研究项目可以描述如下：经典的结果和数论，组合数学，图论，组合博弈论和理论计算机科学的最新发展是通过应用MEX算法和生成互补集获得的。最近PI引入了一种新的MEX算法的推广，令人惊讶的是，应用MEX算法来生成Beatty互补序列相当于将数字前置到无理数的连分数上。PI还表明，迭代MEX算法会产生一个动力系统，其轨道具有二次无理数的参数族不动点。这些结果自然导致的性质，偶数和奇数的正整数是不变的，当应用MEX生成互补序列。这个项目的目标是研究互补序列，MEX算法，连分数和动力系统之间的这些联系，将它们扩展到一般的互补序列，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，从分析和加法数论，以下策略已经采用的PI。这个奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。