Product of Sets on Varieties in Finite Fields

IF 1.2 3区数学 Q2 MATHEMATICS, APPLIED

Journal of Fourier Analysis and Applications Pub Date : 2024-03-27 DOI:10.1007/s00041-024-10079-x

引用次数: 0

Abstract

Let V be a variety in \(\mathbb {F}_q^d\) and \(E\subset V\) . It is known that if any line passing through the origin contains a bounded number of points from E, then \(\left| \prod (E) \right| =|\{x\cdot y:x, y\in E\}|\gg q\) whenever \(|E|\gg q^{\frac{d}{2}}\) . In this paper, we show that the barrier \(\frac{d}{2}\) can be broken when V is a paraboloid in some specific dimensions. The main novelty in our approach is to link this question to the distance problem in one lower dimensional vector space, allowing us to use recent developments in this area to obtain improvements.

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有限域中变量上的集合积

Abstract Let V be a variety in \(\mathbb {F}_q^d\) and \(E\subset V\) .众所周知，如果任何经过原点的直线包含来自 E 的有界数的点，那么只要 \(|E|\gg q^{frac{d}{2}}\) ， \(\left| \prod (E) \right| =|{x\cdot y:x, y\in E\}|\gg q\) .在本文中，我们证明了当 V 在某些特定维度上是抛物面时，障碍 \(\frac{d}{2}\) 可以被打破。我们方法的主要新颖之处在于将这一问题与一个低维向量空间中的距离问题联系起来，使我们能够利用这一领域的最新发展来获得改进。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Fourier Analysis and Applications 数学-应用数学

CiteScore

2.10

自引率

16.70%

发文量

审稿时长

6-12 weeks

期刊介绍： The Journal of Fourier Analysis and Applications will publish results in Fourier analysis, as well as applicable mathematics having a significant Fourier analytic component. Appropriate manuscripts at the highest research level will be accepted for publication. Because of the extensive, intricate, and fundamental relationship between Fourier analysis and so many other subjects, selected and readable surveys will also be published. These surveys will include historical articles, research tutorials, and expositions of specific topics. TheJournal of Fourier Analysis and Applications will provide a perspective and means for centralizing and disseminating new information from the vantage point of Fourier analysis. The breadth of Fourier analysis and diversity of its applicability require that each paper should contain a clear and motivated introduction, which is accessible to all of our readers. Areas of applications include the following: antenna theory * crystallography * fast algorithms * Gabor theory and applications * image processing * number theory * optics * partial differential equations * prediction theory * radar applications * sampling theory * spectral estimation * speech processing * stochastic processes * time-frequency analysis * time series * tomography * turbulence * uncertainty principles * wavelet theory and applications