RUI: Inverse Problems for Finite and Infinite Sets, and Nonstandard Methods

逆问题研究自然数集合A的结构，当A加A（A的两个副本之和）的大小相对较小时。在20世纪50年代末和60年代初，G。A. Freiman导出了一系列定理，表明对于有限集合A，如果A加A的大小很小，则A必须具有某种算术结构。在该提案中，主要研究者提出，研究逆现象的推广弗赖曼定理有限集和发现新的定理无限集使用非标准的方法。Freiman的一种结果最优地刻画了当A加A的大小小于3倍A减2的大小时A的结构。从那时起，许多努力已作出各种人推广弗赖曼定理与这种类型的条件。然而，直到最近，对于A加A的大小大于A减2的大小的3倍，还没有获得A的最佳结构表征。出乎意料的是，非标准方法被引入并取得了突破。在非标准分析的帮助下，首席研究员能够给出A的结构的最佳表征，当A加A的大小上界为c乘以A的大小时，c略大于3。首席研究员认为，这些方法的潜在力量还没有完全被发现。他建议进一步研究，当A加A的大小小于A的三分之一或更大时，对A的结构得出同样的表征。主要研究者还建议使用非标准方法研究其他问题，包括无穷集的逆问题。Freiman的逆现象揭示了自然数的基本行为。该思想在各种领域中有许多应用，例如组合数论、代数、编码理论、整数规划、概率等，如在“Set Addition的Structure Theory”，Asterisque No.258（1999），Societe Mathematique de France，巴黎中所提及的。对Freiman定理的任何改进，必将在这些领域中带来更好的应用.由于编号和计数是数学和我们日常生活的基础，因此更好地了解我们的数字系统显然对两者都有好处。该提案的一个显著特点是使用了非标准方法。非标准分析使用逻辑中的技术来创建一个允许无限大的数字的世界。这项研究的一个有趣的方面是，这些无穷大的数字，这是纯粹想象的人类头脑，可以用来证明定理在真实的世界。逻辑和数论之间的这种相互作用表明了跨学科研究的重要性。这项研究也应该对年轻的研究人员有启发性，鼓励他们开放思想，接受不同种类的知识，并揭示这些不同学科之间隐藏的关系。

Inverse problems study the structure of a set A of natural numbers when the size of A plus A (the sum of two copies of A) is relatively small. During the late 1950's and early 1960's G. A. Freiman derived a series of theorems, which indicated that for a finite set A, if the size of A plus A is small, then A must have some arithmetic structure. In the proposal the principal investigator proposes to study the inverse phenomenon for generalizing Freiman's theorems for finite sets and discovering new theorems for infinite sets using nonstandard methods. One type of Freiman's results optimally characterizes the structure of A when the size of A plus A is less than 3 times the size of A minus 2. Since then many efforts have been made by various people to generalize Freiman's theorems with this type of condition. However, no optimal structural characterization of A had been obtained for the size of A plus A greater then 3 times the size of A minus 2, until recently. It seemed unexpected that nonstandard methods were brought in and made a break-through. With the help of nonstandard analysis the principal investigator was able to give an optimal characterization of the structure of A when the size of A plus A is upper bounded by c times the size of A for a constant c slightly greater than 3. The principal investigator believes that the potential power of the methods hasn't been fully uncovered. He proposes to further his investigation and derive the same kind of characterization for the structure of A when the size of A plus A is less then the ten-thirds the size of A or further. The principal investigator also proposes to work on other problems including inverse problems for infinite sets using the nonstandard methods.Freiman's inverse phenomenon revealed a fundamental behavior of natural numbers. This idea had many applications in various fields such as combinatorial number theory, algebra, coding theory, integer programming, probability, etc. as mentioned in "Structure Theory of Set Addition", Asterisque No. 258 (1999), Societe Mathematique de France, Paris. Any improvement on Freiman's theorems would certainly bring about better applications in these fields. Since numbering and counting are fundamental to mathematics as well as our daily lives, a better understanding of our number system would obviously be beneficial to the both. One distinctive feature of this proposal is the use of the nonstandard methods. Nonstandard analysis uses the techniques in logic to create a world in which infinitely large numbers are allowed. An interesting aspect of this research is that these infinitely large numbers, which are purely imaginary to the human minds, could be used to prove theorems in the real world. This interaction between logic and number theory demonstrates the importance of interdisciplinary research. This research should also be instructive to young researchers encouraging them to open their minds, embrace different kinds of knowledge, and uncover the hidden relationships between these different disciplines.