p-adic Ghobber-Jaming Uncertainty Principle

TL;DR

This paper establishes a p-adic version of the Ghobber-Jaming Uncertainty Principle, providing bounds on vector norms in finite-dimensional p-adic Hilbert spaces based on orthonormal bases and inner product constraints.

Contribution

It introduces the p-adic Ghobber-Jaming Uncertainty Principle, extending the classical uncertainty principle to p-adic and non-Archimedean Banach space settings.

Findings

Derived a p-adic uncertainty inequality for finite-dimensional Hilbert spaces.

Extended the inequality to non-Archimedean Banach spaces.

Provided bounds relating inner products and vector norms in p-adic contexts.

Abstract

Let $\{\tau_j\}_{j=1}^n$ and $\{\omega_k\}_{k=1}^n$ be two orthonormal bases for a finite dimensional p-adic Hilbert space $\mathcal{X}$. Let $M,N\subseteq \{1, \dots, n\}$ be such that \begin{align*} \displaystyle \max_{j \in M, k \in N}|\langle \tau_j, \omega_k \rangle|<1, \end{align*} where $o(M)$ is the cardinality of $M$. Then for all $x \in \mathcal{X}$, we show that \begin{align} (1) \quad \quad \quad \quad \|x\|\leq \left(\frac{1}{1-\displaystyle \max_{j \in M, k \in N}|\langle \tau_j, \omega_k \rangle|}\right)\max\left\{\displaystyle \max_{j \in M^c}|\langle x, \tau_j\rangle |, \displaystyle \max_{k \in N^c}|\langle x, \omega_k\rangle |\right\}. \end{align} We call Inequality (1) as \textbf{p-adic Ghobber-Jaming Uncertainty Principle}. Inequality (1) is the p-adic version of uncertainty principle obtained by Ghobber and Jaming \textit{[Linear Algebra Appl., 2011]}. We also derive analogues of Inequality (1) for non-Archimedean Banach spaces.