Sidon sets and 2-caps in 𝔽3n

Huang, Yixuan; Tait, Michael; Won, Robert

doi:10.2140/involve.2019.12.995

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Sidon sets and 2-caps in $\mathbb{F}_3^n$

Yixuan Huang, Michael Tait and Robert Won

Vol. 12 (2019), No. 6, 995–1003

DOI: 10.2140/involve.2019.12.995

Abstract

For each natural number $d$ , we introduce the concept of a $d$ -cap in $F_{3}^{n}$ . A set of points in $F_{3}^{n}$ is called a $d$ -cap if, for each $k = 1, 2, \dots, d$ , no $k + 2$ of the points lie on a $k$ -dimensional flat. This generalizes the notion of a cap in $F_{3}^{n}$ . We prove that the $2$ -caps in $F_{3}^{n}$ are exactly the Sidon sets in $F_{3}^{n}$ and study the problem of determining the size of the largest $2$ -cap in $F_{3}^{n}$ .

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Keywords

Sidon sets, cap sets, caps, 2-caps

Mathematical Subject Classification 2010

Primary: 05B10, 05B25, 05B40, 51E15

Milestones

Received: 16 September 2018

Revised: 7 February 2019

Accepted: 18 February 2019

Published: 3 August 2019

Communicated by Joshua Cooper

Authors

Yixuan Huang
Department of Mathematics and Statistics Wake Forest University Winston-Salem, NC United States

Michael Tait
Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA United States

Robert Won
Department of Mathematics University of Washington Seattle, WA United States

Vol. 12, No. 6, 2019