Arithmetic combinatorics and applications to number theory

算术组合及其在数论中的应用

• 批准号: 1301608
• 负责人: Mei-Chu Chang
• 金额: $ 17.55万
• 依托单位: University of California-Riverside
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-01 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1301608&HistoricalAwards=false
• 关键词: Arithmetic; combinatorics; applications; number; theory

This is a proposal in arithmetic combinatorics, which has become an interdisciplinary field of research with many applications. While certain themes in additive combinatorics are classical in number theory, there is also focus on new structural questions that turned out to be important. For instance, the work of Gowers and, later, Green and Tao on arithmetic progressions have put considerable impetus on Freiman's theorem and it's quantitative versions. Parallel to these results, a general `sum-product' theory in finite fields and residue rings was developed. The roots of this research go back for instance to early work of Erdos-Szemeredi and the finite field version of the Kakeya problem (solved by Z. Dvir). It turned out that results from sum-product and product theory in various settings are of interest in their own right as they lead to new results in analytic number theory (such as estimates of short character sums), in pseudo-randomness and in group theory (growth, expansion and spectral gaps). The PI intends to explore finite field analogues of the Szemeredi Trotter theorem on incidences for algebraic curves (which are a special case of so-called pseudo-line systems). In the finite field setting, such results are presently only available for straight lines. More general results for pseudo-line systems have been obtained over the reals but none of the known approaches seem adaptable to the finite fields situation, so that new ideas are clearly needed here. Results of this type would have major implications in the areas mentioned above because they allow to obtain nontrivial statements on solutions of systems of equations in situations where classical number theory is not applicable. As a particular case, the PI would like to obtain analogues of the Bombieri-Pila results on lattice points on curves restricted to boxes in the prime field setting. Related to 'growth' phenomena in groups, the work of Breuillard, Green and Tao provides a complete description of 'approximate groups' that in some sense generalize Freiman's theorem and also provide a finitary version of Gromov's theorem. At this stage, the results are only qualitative and obtaining quantitative versions would be most interesting, in particular in view of the consequences to group expansion. The PI will continue to work with her collaborators on Poonen's conjecture on the multiplicative order of F-points on curves. Again this is a problem at the interface of combinatorics, algebra and number theory where progress can be expected.Arithmetic combinatorics and `sum-product theory' in various settings have become increasingly significant to various other fields, such as pseudo-randomness in computer science, classical analytic number theory and the theory of expansion in linear groups. The purpose of this proposal is to continue research on the related issues in combinatorial number theory and their applications in particular, to problems of estimating the number of solutions of algebraic equations when the variables are restricted. This research involves different groups of people and the interaction of various branches of mathematics, occasionally leading to progress on old problems.

这是算术组合学中的一个建议，算术组合学已经成为一个具有许多应用的跨学科研究领域。虽然加法组合学中的某些主题在数论中是经典的，但也有一些新的结构问题被证明是重要的。例如，高尔斯以及后来的格林和陶在算术级数方面的工作对弗莱曼定理及其量化版本产生了相当大的推动作用。与这些结果平行的是，在有限域和剩余环上发展了一个一般的“和积”理论。例如，这项研究的根源可以追溯到鄂尔多斯-斯梅雷迪的早期工作和有限域版本的Kakeya问题(由Z.Dvir解决)。结果表明，在不同的环境下，和积论和乘积理论的结果本身是有意义的，因为它们导致了解析数理论(如短特征和的估计)、伪随机性和群论(增长、扩展和谱间隙)的新结果。PI打算探索关于代数曲线(这是所谓伪直线系统的特例)的关联的Szmeredi Trotter定理的有限域类似。在有限域中，这样的结果目前只适用于直线。关于伪线系统的更一般的结果已在实数上得到，但已知的方法似乎都不适用于有限域的情况，因此这里显然需要新的想法。这种类型的结果将在上述领域具有重大意义，因为它们允许在经典数论不适用的情况下获得关于方程组解的非平凡陈述。作为一种特殊情况，PI希望得到与Bombieri-Pila结果类似的结果，该结果关于仅限于素数域设置中的框的曲线上的格点。与群中的“增长”现象有关，Breuillard，Green和Tao的工作提供了对“近似群”的完整描述，在某种意义上推广了Freiman定理，也提供了Gromov定理的有限版本。在这个阶段，结果只是定性的，获得定量的版本将是最有趣的，特别是考虑到对集团扩张的影响。PI将继续与她的合作者一起研究Poonen关于曲线上F点的乘法阶数的猜想。同样，这是一个在组合学、代数和数论的交界处可以期待进步的问题。各种背景下的数学组合学和‘和积论’对其他各种领域变得越来越重要，例如计算机科学中的伪随机性、经典解析数论和线性群中的展开理论。本文的目的是继续研究组合数论及其应用中的相关问题，特别是在变量受限时估计代数方程解的个数问题。这项研究涉及不同的人群和不同数学分支的相互作用，偶尔会在老问题上取得进展。