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用傅里叶变换的波的叠加来表述。一个基本的问题是“这些波的频率告诉我们什么关于他们的干扰，反之亦然？".波的干涉与它们叠加的幅度有关，并且可以通过根据严格的模式选择频率来增大干涉的集中度。PI所追求的问题之一是匡威的：这些模式是实现非常集中的叠加所必需的吗？这就是所谓的逆Littlewood问题。第二个正在研究的问题是不确定性原理。假设我们有两个波族可以用来分解一个信号，而这两个波族是非常不相容的。这种不相容性意味着我们可以预期，来自一个族的简单波看起来要复杂得多，因为它是来自第二个族的叠加波。一个具体的例子是，一个乘法特征（一个族中非常简单的波）是否可以分解为一个加法卷积（另一个族中相当简单的叠加）。在数论中比较序列的分布和算术结构有很长的历史。传统上，通过傅里叶变换观察到这种联系。为了更好地理解这一现象，PI计划建立在最近的结果上，这些结果将格的有限子集的维数与其相关傅立叶级数的L^1范数估计联系起来。这一领域的研究始于利特尔伍德关于有限整数集合的傅里叶变换的L^1 -范数可以有多小的猜想。这个问题在1980年代由Konyagin和McGehee-Pigno-Smith独立解决。最近，问题已经转向对这个问题中的极端化进行分类，PI最近证明了它们在适当的意义上必须是低维的。在这个项目中，他将进一步发展该方法，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。在有限域上，PI和Petridis在Sarkozy关于和集和二次剩余的问题上取得了实质性的进展。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，以便认真解决这个猜想。除此之外，他还将使该方法适用于其他问题，这反过来又将导致维诺格拉多夫关于最小非剩余的猜想取得进展。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。