Analytic Number Theory Motivated by Approximate Translation Invariance

指数和是编码算术信息的傅立叶级数。这些和的点边界和平均值在整个解析数论中起着基本的作用，并且是检验序列的均匀分布（明显的“随机性”）的主要工具，它支撑着数论在理论计算机科学、密码学等领域的许多应用。直到最近十年，尽管从Hardy和Littlewood引入他们著名的圆法开始，近一个世纪的紧张努力，关于多项式的指数和的平均值的主要猜想仍然没有解决，除了最简单的涉及线性和二次多项式的情况。在过去的五年里，十年的戏剧性进展达到了顶峰，证明了最雄心勃勃的猜想，这些猜想是关于这些均值猜想的一个中心例子，与维诺格拉多夫的均值定理有关，一方面由布尔甘、德墨忒尔和古斯通过解耦，另一方面由提议者通过嵌套有效同余。在这个项目中，首席研究员将加强、扩展和利用这些最新的方法，以便在定量算术几何和更广泛的Hardy-Littlewood方法理论中应用的一系列均值猜想中获得类似的决定性进展。这将有助于完全一般性地解决许多变量的平移-膨胀不变系统的主要猜想，包括涉及当前技术无法达到的小尺度集平均值的此类猜想的类似物。研究生将在这一重要的新兴领域进行培训，并且提议者将编写一篇新的文本，旨在介绍翻译-扩张不变系统的有效同余，作为介绍有理数、数域和函数域上的圆方法的现代发展的工具。最近在对指数和均值的理解方面取得的进展已经提出了维诺格拉多夫均值的主要猜想。通过正交性，该平均值与平移-膨胀不变丢番图系统相关联。尽管取得了这一成功，但解耦方法和嵌套有效同余方法目前都不能解决多变量平移-扩张不变系统的萌芽问题。此外，它们没有处理单位超立方体子集上支持的相应平均值，因此无法为Hardy-Littlewood方法中使用的小弧或大弧集提供有用的估计。本项目将在这一综合理论上取得决定性的进展，提供有关指数和平均值的主要猜想，不仅与一般平移-膨胀不变丢番图系统有关，而且与只具有部分或近似平移-膨胀不变结构的系统有关。这将通过采用提议者的嵌套有效的同余方法，在相当一般的数字字段和函数字段设置中完成。这种灵活的方法集允许在多齐次设置中将同余信息从一组变量传递到另一组变量，甚至在处理有限的积分域时也可以实现这一点。在这些新估计的应用中，主要研究者将通过Hardy-Littlewood方法在具有对角结构的超曲面上建立具有有理系数的有理曲线存在的局部-全局原理。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.