Arithmetic structure and distribution

许多算术问题都可以用傅里叶变换的方式来表达波的叠加。一个基本的问题是“这些波的频率告诉我们它们的干扰是什么，反之亦然？”波的干涉与它们叠加的大小有关，根据严格的模式选择频率可以使干涉的集中程度变大。PI所追求的问题之一是相反的：这些模式是实现非常集中的叠加所必需的吗？这叫做逆利特尔伍德问题。调查的第二个问题是不确定性原则。假设我们有两个波族可以用来分解一个信号，这两个波族是非常不相容的。这种不相容意味着我们期望来自一个家族的简单波作为来自第二个家族的叠加看起来要复杂得多。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性卷积（另一个族中相当简单的叠加）。这就是所谓的“萨科齐问题”。数论中数列的分布和算术结构的比较由来已久。经典地，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，将晶格的有限子集的维度与其相关傅立叶级数的L^1范数的估计联系起来。这个领域的研究开始于Littlewood关于有限整数的傅里叶变换的L^1范数可以有多小的猜想。这个问题在20世纪80年代由Konyagin和mcgee - pigno - smith独立解决。最近，问题转向了对这个问题中的极值值进行分类，PI最近证明了它们必须在适当的意义上是低维的。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。在有限域环境下，PI和Petridis在关于集和二次残的Sarkozy问题上取得了实质性的进展。他们解决了几乎所有质数的猜想。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，他将把该方法应用于其他问题，这反过来又将导致维诺格拉多夫关于最小非残数猜想的进展。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。

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.