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

逆问题研究的是当a + a （a的两个副本的和）的大小相对较小时，自然数集合a的结构。在20世纪50年代末和60年代初，G. a . Freiman推导了一系列定理，这些定理表明，对于有限集合a，如果a + a的大小很小，那么a一定具有某种算术结构。在提案中，主要研究者提出了利用非标准方法研究有限集上推广Freiman定理的逆现象和发现无限集上新定理的方法。当A + A的大小小于A - 2的大小的3倍时，一类Freiman的结果最优地表征了A的结构。从那时起，各种各样的人做出了许多努力，用这种类型的条件推广了弗莱曼定理。然而，直到最近，对于A + A的尺寸大于A - 2的3倍的尺寸，还没有得到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的最佳结构表征，当A + A的大小上限为c乘以A的大小，常数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。他建议进一步研究并推导出当A + A的大小小于A的三分之十或更大时A的结构的相同特征。作者还提出了利用非标准方法研究无限集逆问题等其他问题的方法。弗莱曼的逆现象揭示了自然数的一个基本行为。这一思想在组合数论、代数、编码理论、整数规划、概率论等各个领域都有很多应用，如《Set Addition的结构理论》，Asterisque No. 258(1999)，法国数学协会，巴黎。对弗莱曼定理的任何改进都必将在这些领域带来更好的应用。由于数数和计数是数学和我们日常生活的基础，更好地了解我们的数字系统显然对两者都有好处。这一建议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造一个允许无限大数字的世界。这项研究的一个有趣的方面是，这些对人类头脑来说纯粹是想象的无极大数，可以用来证明现实世界中的定理。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指导意义，鼓励他们开放思想，拥抱不同的知识，揭示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.