Sparse Bounds and Improving Estimates, Continuous and Discrete

平均总是使函数或数据变得平滑或改进。在连续的情况下，这种现象已经被深入研究了几十年。如果我们对离散环境中的对象进行平均，比如整数，或者其他一些高维晶格，平均的平滑特性直到最近才开始被研究。一个自然要研究的对象是五维晶格中离散球面上的平均值。证明平均值能改善函数涉及数论和分析的一系列深层次问题。然而，解决这个问题的自然极限似乎仍然很困难。在这个项目中，这些问题将在涉及连续和离散现象的环境中进行探讨。这些问题也都是初级的，这使得它们适合指导项目，从本科到博士后水平。自20世纪70年代以来，人们对连续环境中不断改善的不平等现象进行了广泛研究。然而，离散环境下的相应问题才刚刚引起人们的注意。例如，d维整数在离散球面上的平均值，d至少为5，具有丰富的改进不等式的理论，与更广为人知的连续情况相平行。然而，这些证明涉及到多频分析带来的复杂性，以及对Kloosterman和的精细估计。还有一个更丰富的理论，它的轮廓正在出现，给出了一些算术变种的尖锐范围的不等式。这个主题也允许某些稀疏界的证明。后者是不断改善的不平等的无标度版本。稀疏界立即意味着其他加权和向量值的结果。后者在这门学科中是新的。这些问题揭示了这些平均算子的新方面，并需要新的研究模式，深化调和分析与数论之间的联系。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。

Averaging always smooths out, or improves, functions or data. In the continuous case, this phenomena has been intensively studied for decades. If we average over objects in a discrete setting like the integers, or some other high dimensional lattice, the smoothing properties of the average have only recently started to be studied. A natural object to study is the average over a discrete sphere in a five dimensional lattice. The proof that the average improves functions engages a range of deep aspects within number theory and analysis. However, resolving the natural limits for this question still seem difficult. In this project, these questions will be explored in a setting that involves both continuous and discrete phenomena. These questions are also elementary to state, which makes them amenable for mentoring programs, from undergraduate through postdoctoral levels. An improving inequality in the continuous setting has been widely studied since the 1970's. However, the corresponding questions in the discrete setting have only just attracted attention. For instance, averages over the discrete sphere in the d-dimensional integers, with d at least 5, have a rich theory of improving inequalities, paralleling the much better known continuous case. The proofs however involve complications arising from multi-frequency analysis, as well as fine estimates on Kloosterman sums. There is a richer theory, the outlines of which are appearing, giving a sharp range of inequalities for some arithmetic varieties. This subject also allows for the proof of certain sparse bounds. The latter are scale-free versions of the improving inequalities. A sparse bound immediately implies other weighted and vector valued consequences. The latter are new in this subject. These questions reveal new aspects of these averaging operators, and require new modes of investigation, deepening the connection between harmonic analysis and number theory.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.