The VC dimension of quadratic residues in finite fields

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics Pub Date : 2024-08-16 DOI:10.1016/j.disc.2024.114192

引用次数: 0

Abstract

We study the Vapnik–Chervonenkis (VC) dimension of the set of quadratic residues (i.e. squares) in finite fields, $F_{q}$ , when considered as a subset of the additive group. We conjecture that as $q \to \infty$ , the squares have the maximum possible VC-dimension, viz. $(1 + o (1)) \log_{2} q$ . We prove, using the Weil bound for multiplicative character sums, that the VC-dimension is $⩾ (\frac{1}{2} + o (1)) \log_{2} q$ . We also provide numerical evidence for our conjectures. The results generalize to multiplicative subgroups $Γ \subseteq F_{q}^{\times}$ of bounded index.

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有限域中二次残差的 VC 维数

我们研究了有限域 Fq 中二次残差（即正方形）集合作为加法群子集时的 Vapnik-Chervonenkis （VC）维度。我们猜想，当 q→∞ 时，正方形具有最大可能的 VC 维度，即 (1+o(1))log2q。我们利用乘法特征和的韦尔界证明，VC 维度为⩾(12+o(1))log2q。我们还为我们的猜想提供了数字证据。这些结果可以推广到有界指数的乘法子群 Γ⊆Fq×。

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

Discrete Mathematics 数学-数学

CiteScore

1.50

自引率

12.50%

发文量

424

审稿时长

6 months

期刊介绍： Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.