$p$-adic equiangular lines and $p$-adic van Lint–Seidel relative bound
Bulletin Polish Acad. Sci. Math.
MSC: Primary 12J25; Secondary 46S10, 47S10, 11D88
DOI: 10.4064/ba240825-18-10
Published online: 31 October 2024
Abstract
We introduce the notion of $p$-adic equiangular lines and derive the first fundamental relation between common angle, dimension of the space and the number of lines. More precisely, we show that if $\{\tau _j\}_{j=1}^n$ is a collection of $p$-adic $\gamma $-equiangular lines in $\mathbb Q^d_p$, then $$|n|^2\leq |d|\max \,\{|n|, \gamma ^2 \}.$$ We call this inequality the $p$-adic van Lint–Seidel relative bound. We believe that this complements the fundamental van Lint–Seidel [Indag. Math. 28 (1966)] relative bound for equiangular lines in the $p$-adic case.
