Geometric and algebraic methods in Erdos type problems

鄂尔多斯型问题的几何与代数方法

• 批准号: RGPIN-2018-03880
• 负责人: Solymosi, Jozsef
• 金额: $ 5.97万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758582
• 关键词: Geometric; algebraic; methods; Erdos; type

The main objective of the proposed research program is to develop new and improved techniques to attack arithmetic problems in discrete geometry and additive combinatorics.Several central problems in different parts of mathematics can be translated into questions in discrete geometry. In many cases, such connections were discovered relatively recently due to pioneering works of great mathematicians like Jean Bourgain, Tim Gowers, and Terry Tao. As it turned out there are classical problems in discrete geometry which have direct impacts in harmonic analysis, combinatorics, number theory, and theoretical computer science. This field even has its unique Mathematics Subject Classification code: 52C10, Erdos problems and related topics of discrete geometry. Many of these problems can be formulated as counting (bounding) the maximum number of incidences between lines/curves/planes/surfaces and points. (We say that a point is incident to a line (or curve or surface) if the point lies on the line.) Incidence bounds provide inside information about the arithmetic-geometric structure of the underlying field. A well known example - and an important part of my research - is the sum-product problem: given a finite set of integers, A, is it possible that both the sumset, A+A, and the product set, A*A, are small? (The sumset and product set are the set of pairwise sums and products of elements from A.) For example if A is the set of the first n natural numbers then the sumset is small, it has cardinality 2n-1, while the product set is almost quadratic in n=|A|. If A is a geometric progression then the product set is small, but then the sumset is quadratic in |A|. Erdos and Szemeredi is conjectured that |A+A|+|A*A|>|A|2-epsilon, where epsilon goes to zero as |A| goes to infinity. All significant improvements in this problem have come from (discrete) geometry, by understanding the connections between the geometry of the plane and the arithmetic of the underlying field. The particular problem I will focus on is Erdos' Unit Distances Problem: What is the maximum number of unit distances among pairs of n points on the plane? Erdos conjectured that the upper bound on unit distances is n1+epsilon, where epsilon goes to zero as n goes to infinity. This is a 70 year old problem in which the best upper bound, n4/3, was given more than 30 years ago. I plan to improve this upper bound. There are examples of metrics similar to the Euclidean, where the number of unit distances is n4/3 , so any possible improvement should use more than the combinatorics of unit circle arrangements. Improving this bound might seem to be overly ambitious, but recent developments in using algebraic methods to tackle similar problems make the plan look more feasible.

拟议的研究计划的主要目的是开发新的和改进的技术，以攻击离散几何学和添加剂组合中的算术问题。数学不同部分的几个中心问题可以将其转化为离散几何形状中的问题。在许多情况下，由于Jean Bourgain，Tim Gowers和Terry Tao等伟大的数学家的开创性作品，最近发现了这种联系。事实证明，离散的几何形状存在经典问题，这些问题在谐波分析，组合学，数理论和理论计算机科学方面具有直接影响。该领域甚至具有其独特的数学主题分类代码：52C10，ERDOS问题和离散几何学的相关主题。这些问题中的许多可以作为计数（边界）线/曲线/平面/表面和点之间的最大发病率数。 （我们说，如果点位于线上，则一个点是出现在线（或曲线或表面）的。）入射限制提供了有关基础场的算术几何结构的内部信息。一个众所周知的例子 - 这是我研究的重要部分 - 总和问题是：给定有限的整数集，A，以及库集，a+a和乘积集，a*a，很小吗？ （集合和产品集是来自A的成对总和和元素的集合。如果a是几何进程，则产品集很小，但是在| a |中的集合是二次的。 Erdos和szemeredi猜想| a+a |+| a*a |> | a | a | 2- epsilon，其中epsilon占零作为| a |去无穷大。通过了解平面的几何形状与基础场的算术之间的连接，这一问题的所有显着改善都来自（离散的）几何形状。我将重点关注的特定问题是Erdos的单位距离问题：飞机上N点对之间的最大单位距离数是多少？ Erdos猜想单位距离上的上限为N1+Epsilon，epsilon在n进入无穷大时变为零。这是一个70年的问题，其中最好的上限N4/3在30年前被给出了。我计划改善这种上限。有类似于欧几里得的指标的示例，其中单位距离的数量为N4/3，因此任何可能的改进都应使用比单元圆排列的组合使用更多的。改善这种约束似乎过于雄心勃勃，但是使用代数方法来解决类似问题的最新发展使该计划看起来更可行。