Arithmetic structure and distribution

算术结构和分布

• 批准号: 2001622
• 负责人: Brandon Hanson
• 金额: $ 9.54万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001622&HistoricalAwards=false
• 关键词: Arithmetic; structure; distribution

Many questions in arithmetic can be phrased in terms of the superposition of waves by way of the Fourier Transform. A basic question is "What do the frequencies of these waves tell us about their interference and vice versa?". The interference of the waves has to do with the magnitude of their superposition, and the concentration of the interference can be made large by choosing the frequencies according to a strict pattern. One of the questions pursued by the PI is the converse: are these patterns necessary to achieve a very concentrated superposition. This is called the Inverse Littlewood Problem. The second question under investigation is an uncertainty principle. Suppose we have two families of waves that can be used to decompose a signal, and these two families are very incompatible. This incompatibility means we expect that a simple wave from one family looks much more complicated as a superposition from the second. A specific instance of this is whether a multiplicative character (a very simple wave in one family) can be decomposed as an additive convolution (a fairly simple superposition in another family). This is called Sarkozy's Problem.There is a long history of comparing the distribution and arithmetic structure of sequences in number theory. Classically, the connection is observed by way of the Fourier transform. Towards a better understanding of this phenomenon, the PI plans to build upon recent results that relate the dimension of a finite subset of a lattice to estimates on the L^1-norm of its associated Fourier series. This area of investigation began with a conjecture of Littlewood on how small the L^1 -norm of the Fourier transform of a finite set of integers could be. The problem was resolved independently by Konyagin and McGehee-Pigno-Smith in the 1980s. Recently, questions have turned to classifying the extremizers in this problem, and the PI has recently proved that they must be, in an appropriate sense, low-dimensional. In this project he will develop the method further, by weakening hypotheses and applying these results to problems in arithmetic reliant the L^1 norm. In the finite field setting, the PI and Petridis have made substantial progress on a problem of Sarkozy concerning sumsets and the quadratic residues. They have resolved the conjecture for almost all primes. The PI will develop the ideas further so as to resolve the conjecture in earnest. Beyond this, he will adapt the method to other problems, which in turn will lead to progress on Vinogradov’s conjecture on the least non-residue.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.

算术中的许多问题都可以通过傅立叶变换用波的叠加来表达。一个基本问题是“这些波的​​频率告诉我们它们的干扰，反之亦然？”。波的干扰与其叠加的幅度有关，并且可以通过根据严格的模式选择频率来使干扰的集中度变大。 PI 追求的问题之一恰恰相反：这些模式对于实现非常集中的叠加是否是必要的。这称为逆利特伍德问题。研究的第二个问题是不确定性原理。假设我们有两个可用于分解信号的波族，而这两个波族非常不兼容。这种不兼容性意味着我们预计来自一个家庭的简单波与来自第二个家庭的叠加看起来要复杂得多。一个具体的例子是乘法特征（一个族中非常简单的波）是否可以分解为加法卷积（另一个族中相当简单的叠加）。这被称为萨科齐问题。在数论中比较序列的分布和算术结构有着悠久的历史。传统上，这种联系是通过傅里叶变换来观察的。为了更好地理解这一现象，PI 计划以最近的结果为基础，将晶格有限子集的维数与其相关傅立叶级数的 L^1 范数的估计联系起来。这一领域的研究始于 Littlewood 对有限整数集的傅里叶变换的 L^1 范数可以有多小的猜想。该问题由 Konyagin 和 McGehee-Pigno-Smith 在 20 世纪 80 年代独立解决。最近，问题转向对这个问题中的极端者进行分类，PI 最近证明，在适当的意义上，它们一定是低维的。在这个项目中，他将通过削弱假设并将这些结果应用于依赖 L^1 范数的算术问题来进一步开发该方法。在有限域环境下，PI和Petridis在Sarkozy有关求和集和二次留数的问题上取得了实质性进展。他们解决了几乎所有素数的猜想。 PI将进一步发展这些想法，以真正解决这个猜想。除此之外，他还将将该方法应用于其他问题，这反过来将导致维诺格拉多夫的最小无残留猜想取得进展。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。