Arithmetic combinatorics and applications to number theory

这是算术组合中的一个建议，它已经成为一个具有许多应用的跨学科研究领域。虽然加性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，极大地推动了弗莱曼定理及其定量版本的发展。在此基础上，提出了有限域和剩馀环的“和积”理论。这项研究的根源可以追溯到Erdos-Szemeredi的早期工作和Kakeya问题的有限域版本（由Z. Dvir解决）。事实证明，和积理论和乘积理论在各种情况下的结果本身就很有趣，因为它们在解析数论（如短字符和的估计）、伪随机性和群论（增长、膨胀和谱间隙）中产生了新的结果。PI打算探索Szemeredi Trotter定理在代数曲线（这是所谓伪直线系统的一种特殊情况）的关联上的有限域类似物。在有限域的情况下，这样的结果目前只适用于直线。在实数上得到了伪直线系统的更一般的结果，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思想。这种类型的结果将在上述领域具有重要意义，因为它们允许在经典数论不适用的情况下获得方程组解的非平凡陈述。作为一种特殊情况，PI希望在素数场设置中，在限制于盒子的曲线上获得类似Bombieri-Pila结果的点阵点。与群中的“增长”现象相关，Breuillard， Green和Tao的工作提供了“近似群”的完整描述，在某种意义上推广了Freiman定理，并提供了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，特别是考虑到群体扩张的后果。PI将继续与她的合作者一起研究Poonen关于曲线上f点的乘法阶的猜想。这又是一个结合了组合学，代数和数论的问题。各种情况下的算术组合和“和积理论”在其他领域变得越来越重要，比如计算机科学中的伪随机性、经典解析数论和线性群展开理论。本课题的目的是继续研究组合数论中的相关问题及其应用，特别是在变量受限时代数方程解的个数估计问题。这项研究涉及不同群体的人以及数学各个分支的相互作用，偶尔会在老问题上取得进展。

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.