The VC dimension of quadratic residues in finite fields

The VC dimension of quadratic residues in finite fields

We study the Vapnik–Chervonenkis (VC) dimension of the set of quadratic residues (i.e. squares) in finite fields, Fq, when considered as a subset of the additive group. We conjecture that as q→∞, the squares have the maximum possible VC-dimension, viz. (1+o(1))log2⁡q. We prove, using the Weil bound...

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Veröffentlicht in:Discrete mathematics 2025-01, Vol.348 (1), p.114192, Article 114192
Hauptverfasser: McDonald, Brian, Sahay, Anurag, Wyman, Emmett L.
Format: Artikel
Sprache:eng
Schlagworte:
Combinatorics
Finite fields
Learning theory
Number theory
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Zusammenfassung:We study the Vapnik–Chervonenkis (VC) dimension of the set of quadratic residues (i.e. squares) in finite fields, Fq, when considered as a subset of the additive group. We conjecture that as q→∞, the squares have the maximum possible VC-dimension, viz. (1+o(1))log2⁡q. We prove, using the Weil bound for multiplicative character sums, that the VC-dimension is ⩾(12+o(1))log2⁡q. We also provide numerical evidence for our conjectures. The results generalize to multiplicative subgroups Γ⊆Fq× of bounded index.
ISSN:0012-365X
DOI:10.1016/j.disc.2024.114192