Product of Sets on Varieties in Finite Fields

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research 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

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.