Geometric and algebraic methods in Erdos type problems

拟议的研究计划的主要目标是开发新的和改进的技术来攻击离散几何和加法组合数学中的算术问题。 数学中不同部分的几个中心问题可以转化为离散几何中的问题。在许多情况下，由于让·布尔甘（Jean Bourgain）、蒂姆·高尔斯（Tim Gowers）和特里·陶（Terry Tao）等伟大数学家的开创性工作，这种联系是最近才被发现的。事实证明，离散几何中的经典问题对调和分析、组合数学、数论和理论计算机科学有直接影响。该领域甚至有其独特的数学学科分类代码：52 C10， 离散几何中的鄂尔多斯问题及相关课题。这些问题中的许多可以公式化为计算（边界）线/曲线/平面/曲面和点之间的最大关联数。(We比如说，如果一个点位于直线（或曲线或曲面）上，则该点与直线（或曲线或曲面）关联。）关联边界提供了关于底层场的算术-几何结构的内部信息。一个著名的例子--也是我研究的一个重要部分--是和积问题：给定一个有限的整数集A，和集A+A和积集A*A是否可能都很小？(The和集和积集是来自A的元素的成对和与乘积的集合。例如，如果A是前n个自然数的集合，那么和集是小的，它的基数是2n-1，而乘积集几乎是n= 2的二次集。|一|.如果A是一个几何级数，那么积集是小的，但是和集是二次的，|一|. Erdos和Szemeredi被告知，|A+A| +| A*A|>|一|2-π，其中π变为零，|一|都是无限的这个问题的所有重大改进都来自（离散）几何，通过理解平面几何和基础场的算术之间的联系。 我将关注的具体问题是鄂尔多斯的单位距离问题：平面上n个点对之间的最大单位距离是多少？Erdos证明单位距离的上限是n1+ n，其中n趋于无穷大时n趋于零。这是一个有70年历史的问题，其中最佳上限n4/3是在30多年前给出的。我打算改进这个上界。有一些类似于欧几里德的度量的例子，其中单位距离的数量是n4/3，所以任何可能的改进都应该使用比单位圆排列的组合学更多的东西。改进这个界限似乎过于雄心勃勃，但最近使用代数方法解决类似问题的发展使该计划看起来更可行。

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.