Analytic Number Theory Motivated by Approximate Translation Invariance

指数和是对算术信息进行编码的傅里叶级数。这些和的点态界限和平均值在整个解析数论中起着基本的作用，并为检验序列的均匀分布(明显的“随机性”)贡献了主要工具，这些工具支撑了数论在理论计算机科学、密码学等方面的许多应用。直到最近十年，尽管从Hardy和Littlewood引入他们著名的圆法开始，经过近一个世纪的紧张努力，关于多项式上指数和的平均值的主要猜想仍然没有解决，除了涉及线性和二次多项式的最简单的情况外。在过去的五年里，十年的戏剧性进展达到了顶峰，证明了与维诺格拉多夫中值定理有关的最雄心勃勃的猜想，这些猜想与维诺格拉多夫的中值定理有关，一方面是由Bourain、Demeter和Guth通过解耦得到的，另一方面是由提出者通过嵌套有效同余的方式证明的。在这个项目中，首席研究人员将增强、扩展和利用这些最新的方法，以便在一系列在定量算术几何和Hardy-Littlewood方法的更广泛理论中应用的均值猜想方面取得类似的决定性进展。这将有助于全面解决许多变量的平移-膨胀不变系统的主要猜想，包括此类猜想的类似猜想，这些猜想涉及在小测量集上平均的平均值，超出了当前技术的范围。一名研究生将在这个重要的新兴领域接受培训，提出者将致力于一本新的文本，旨在介绍平移-膨胀不变系统的有效同余，作为一种工具，介绍有理整数、数域和函数域上圆法的现代发展。在理解指数和的平均值方面的最新进展为Vinogradov的平均值提供了主要猜想。通过正交性，这个平均值与平移-膨胀不变丢番图系联系在一起。尽管取得了这样的成功，但解耦方法和嵌套有效的同余方法目前都不能解决除胚胎多变量平移膨胀不变系统之外的任何系统。此外，它们没有解决单位超立方体的子集上支持的相应平均值，因此无法提供Hardy-Littlewood方法中使用的小圆弧或大圆弧的广泛集合的有用估计。这个项目将在这一综合理论上取得决定性的进展，提出关于指数和平均值的主要猜想，不仅与一般的平移-膨胀不变丢番图系统有关，而且与仅具有部分或近似平移-膨胀不变结构的系统有关。这一切都将在数字字段和函数字段的相当一般的设置中通过采用提出者的嵌套有效同余方法来完成。这一灵活的方法集允许在多同类设置中将一致性信息从一组变量传递到另一组变量，即使在有限的积分域上工作时也可以实现这一点。在这些新估计的应用中，首席研究员将通过Hardy-Littlewood方法建立局部-全局原理，在具有一定对角结构测量的超曲面上存在具有有理系数的有理曲线。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Exponential sums are Fourier series encoding arithmetic information. Pointwise bounds and mean values of such sums play a fundamental role throughout analytic number theory, and contribute the primary tool for testing equidistribution (apparent ``randomness'') of sequences underpinning many applications of number theory in theoretical computer science, cryptography, and so on. Until the last decade, despite almost a century of intense effort starting with the introduction by Hardy and Littlewood of their famous circle method, the main conjectures concerning mean values of exponential sums over polynomials remained unsolved in all but the very simplest cases involving linear and quadratic polynomials. A decade of dramatic progress has culminated in the last five years with the proof of the most ambitious conjectures concerning a central example of such mean value conjectures, that associated with Vinogradov's mean value theorem, on the one hand by Bourgain, Demeter and Guth via decoupling, and on the other by the proposer by means of nested efficient congruencing. In this project, the principal investigator will enhance, extend and exploit these very recent methods so as to obtain similarly decisive progress in an array of mean value conjectures having applications in quantitative arithmetic geometry and the wider theory of the Hardy-Littlewood method. This will contribute to the resolution of the main conjectures for translation-dilation invariant systems in many variables in full generality, including analogues of such conjectures involving mean values averaged over sets of small measure, beyond the reach of current technology. A graduate student will be trained in this important emerging area, and the proposer will work on a new text intended to provide an introduction to efficient congruencing for translation-dilation invariant systems as a vehicle for introducing modern developments in the circle method over the rational integers, number fields and function fields.Very recent advances in the understanding of mean values of exponential sums have delivered the Main Conjecture for Vinogradov's mean value. By orthogonality, this mean value is associated with a translation-dilation invariant Diophantine system. Despite this success, neither the decoupling method nor the nested efficient congruencing method currently address any but embryonic multivariable translation-dilation invariant systems. Moreover, they do not address corresponding mean values supported on subsets of the unit hypercube, and thus fail to provide useful estimates for either minor arcs or wide sets of major arcs of use in the Hardy-Littlewood method. This project will make decisive progress on this comprehensive theory, delivering the main conjectures concerning mean values of exponential sums associated not only with general translation-dilation invariant Diophantine systems, but also systems possessing only partial or approximate translation-dilation invariant structure. This will all be done in the quite general setting of number fields and function fields by adapting the proposer’s nested efficient congruencing methods. This flexible set of methods permits congruence information to be passed from one set of variables to another in multi-homogeneous settings, and this may be achieved even when working on restricted domains of integration. Amongst applications of these new estimates, the principal investigator will establish local-global principles for the existence of rational curves with rational coefficients on hypersurfaces possessing some measure of diagonal structure via the Hardy-Littlewood method.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.