Diophantine Approximation and Aperiodic Order

丢番图近似和非周期阶

• 批准号: 2001248
• 负责人: Alan Haynes
• 金额: $ 18.67万
• 依托单位: University of Houston
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-15 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001248&HistoricalAwards=false
• 关键词: Diophantine; Approximation; Aperiodic; Order

This award aims to develop and harvest a collection of interrelated results spanning the fields of number theory, dynamical systems, and aperiodic order. A central theme is to explore problems which have known applications to the mathematics underpinning models for physical materials called quasicrystals. The research goals are to use tools from number theory, probability, topological dynamics, and ergodic theory, to prove new results about a number of long standing open problems with significance in the world of pure mathematics, which also have the potential for real world applications. The Principal Investigarot will mentor graduate students on topics related to this award.Quasicrystals are physical materials with highly ordered molecular structures, causing them to produce pure point diffraction patterns, but which also possess rotational symmetries that are forbidden by the classical Crystallographic Restriction Theorem. The discovery of these materials in the 1980's by Dan Shechtman revolutionized the world of crystallography and later earned Shechtman a Nobel prize. In the mathematical world the study, and even the existence, of quasicrystals is intimately related to the theory of aperiodic tilings of Euclidean space. Quasicrystals are often modeled by cut and project sets, which give an abundance of examples of such tilings and which, generically, are good examples of systems which possess `aperiodic order'. In the last decade there has been an explosion of insight linking problems in number theory with problems about cut and project sets. A number of unsolved problems in Diophantine approximation have been reformulated, and some have been solved, using ideas from the theory of mathematical quasicrystals. Major problems about sets with aperiodic order have also been solved by understanding the reverse connections. The research in this project is centered around well-known open problems such as the Littlewood Conjecture and the Pisot Conjecture, which are sitting on the border of these subjects. It aims to develop the theory around them in order to create progress in both the realms of pure and applied mathematics.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.

该奖项旨在开发和收获一系列跨越数论，动力系统和非周期性秩序领域的相互关联的结果。一个中心主题是探索已知的应用数学基础模型的物理材料称为准晶体的问题。研究目标是使用数论，概率论，拓扑动力学和遍历理论的工具，证明一些长期存在的开放问题的新结果，这些问题在纯数学领域具有重要意义，也具有真实的世界应用的潜力。准晶体是具有高度有序分子结构的物理材料，可以产生纯点衍射图案，但也具有经典晶体学限制定理所禁止的旋转对称性。20世纪80年代，Dan Shechtman发现了这些材料，彻底改变了晶体学的世界，后来为Shechtman赢得了诺贝尔奖。在数学世界中，准晶的研究，甚至准晶的存在，都与欧几里得空间的非周期镶嵌理论密切相关。准晶通常是由切割和投影集，这给了这样的平铺的例子，一般来说，是很好的例子，具有“非周期秩序”的系统。在过去的十年里，将数论中的问题与割集和投影集问题联系起来的见解激增。丢番图近似中的一些未解决的问题已经被重新表述，有些已经解决，使用数学准晶理论的思想。关于非周期序集合的主要问题也通过理解反向联系得到了解决。该项目的研究围绕着著名的开放问题，如Littlewood猜想和Pisot猜想，这些问题位于这些学科的边缘。该奖项反映了NSF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。

项目成果

Level spacing statistics for the multi-dimensional quantum harmonic oscillator: Algebraic case

多维量子谐振子的能级间距统计：代数情况

• DOI: 10.1063/5.0064523
• 发表时间: 2022
• 期刊: Journal of Mathematical Physics
• 影响因子: 1.3
• 作者: Haynes, Alan;Roeder, Roland
• 通讯作者: Roeder, Roland

A Five Distance Theorem for Kronecker Sequences

克罗内克序列的五距离定理

• DOI: 10.1093/imrn/rnab205
• 发表时间: 2021
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Haynes, Alan;Marklof, Jens
• 通讯作者: Marklof, Jens

Badly approximable points fordiagonal approximation in solenoids

螺线管中对角线近似的不良近似点

• DOI: 10.4064/aa200425-7-12
• 发表时间: 2021
• 期刊: Acta Arithmetica
• 影响因子: 0.7
• 作者: Chen, Huayang;Haynes, Alan
• 通讯作者: Haynes, Alan

Bounded remainder sets for rotations on higher-dimensional adelic tori

高维 adelic tori 上旋转的有界余数集

• DOI: 10.2140/moscow.2021.10.111
• 发表时间: 2021
• 期刊: Moscow Journal of Combinatorics and Number Theory
• 作者: Das, Akshat;Furno, Joanna;Haynes, Alan
• 通讯作者: Haynes, Alan

A three gap theorem for adeles

阿德勒的三间隙定理

• DOI: 10.1007/s11139-022-00648-3
• 发表时间: 2023
• 期刊: The Ramanujan Journal
• 作者: Das, Akshat;Haynes, Alan
• 通讯作者: Haynes, Alan