Geometric and algebraic methods in Erdos type problems

提出的研究计划的主要目标是开发新的和改进的技术来解决离散几何和加性组合中的算术问题。数学不同部分的几个中心问题可以转化为离散几何的问题。在许多情况下，这种联系是最近才被发现的，这是由于像让·布尔甘（Jean Bourgain）、蒂姆·高尔斯（Tim Gowers）和特里·陶（Terry Tao）这样的伟大数学家的开创性工作。事实证明，离散几何中的一些经典问题对谐波分析、组合学、数论和理论计算机科学有直接的影响。该领域甚至有其独特的数学学科分类代码：52C10， Erdos问题和离散几何的相关主题。许多这样的问题可以表述为计算（限定）线/曲线/平面/曲面与点之间的最大重合数。（如果一个点在直线上，我们就说这个点与直线（曲线或曲面）相接。）关联界提供了底层场的算术-几何结构的内部信息。一个众所周知的例子——也是我研究的一个重要部分——是和积问题：给定一个有限整数集A，是否有可能A+A和乘积集A*A都很小？（sumset和product set是a中元素的成对和和和的集合）例如，如果A是前n个自然数的集合，那么sumset很小，它的基数为2n-1，而乘积集在n=|A|中几乎是二次的。如果A是一个几何级数，那么乘积集很小，但是这个乘积集在|A|中是二次的。Erdos和Szemeredi推测|A+A|+|A*A|>|A|2-当|A|趋于无穷时，趋于零。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底层场的算术之间的联系。******我将重点关注的问题是Erdos的单位距离问题：平面上n个点对之间单位距离的最大数目是多少？Erdos推测单位距离的上界是n1+，当n趋于无穷时趋于零。这是一个有70年历史的问题其中的最佳上界，n4/3，是在30多年前给出的。我打算改进这个上界。有一些度量类似于欧几里得的例子，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组合学更多的方法。改善这一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进展使该计划看起来更加可行。**

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. **