Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On an uncertainty principle for small index subgroups of finite fields

Diego Fernando Díaz Padilla, Jesús Alonso Ochoa Arango

TL;DR

This work addresses the nonvanishing minors property for compressed Fourier transforms attached to index-$3$ subgroups $H\le \mathbb{F}_q^{\times}$ with nontrivial characters $\chi$, extending prior results that treated the trivial case. It derives concise necessary and sufficient conditions in terms of Gaussian sums $G_i$ of character extensions and the cubic sums $T_j=\sum_{i=0}^2 \zeta_3^{ji}G_i$, proving that $NVM$ holds iff some $G_i\neq G_j$ and $T_0\neq 0$, with an explicit construction for the index-3 case using orbit representatives and a $3\times3$ compressed Fourier matrix whose determinant is $-27G_0G_1G_2$. The results yield an equivalent Biró–Meshulam–Tao uncertainty principle formulation: for every nonzero $\chi$-symmetric $f$, $|\mathrm{supp}(f)|+|\mathrm{supp}(\hat{f})|\ge q+(q-1)/3-1$ under the same conditions. By leveraging orbit-representative structures, character extensions, and Gaussian-sum identities, the paper broadens the understanding of uncertainty in finite-field transforms beyond the trivial character case and clarifies when NVM can be achieved for index-$3$ subgroups.

Abstract

In this paper we continue the study of the nonvanishing minors property (NVM) initiated by Garcia, Karaali and Katz, for the compressed Fourier matrix attached to a subgroup $H$ of the multiplicative group of a finite field $\mathbb{F}_q$ and a character $χ$ defined over $H$. Here we provide a characterization of this aforementioned property for \textit{symmetries} arising from an index-3 subgroup $H$ and a nontrivial character $χ$.

On an uncertainty principle for small index subgroups of finite fields

TL;DR

This work addresses the nonvanishing minors property for compressed Fourier transforms attached to index- subgroups with nontrivial characters , extending prior results that treated the trivial case. It derives concise necessary and sufficient conditions in terms of Gaussian sums of character extensions and the cubic sums , proving that holds iff some and , with an explicit construction for the index-3 case using orbit representatives and a compressed Fourier matrix whose determinant is . The results yield an equivalent Biró–Meshulam–Tao uncertainty principle formulation: for every nonzero -symmetric , under the same conditions. By leveraging orbit-representative structures, character extensions, and Gaussian-sum identities, the paper broadens the understanding of uncertainty in finite-field transforms beyond the trivial character case and clarifies when NVM can be achieved for index- subgroups.

Abstract

In this paper we continue the study of the nonvanishing minors property (NVM) initiated by Garcia, Karaali and Katz, for the compressed Fourier matrix attached to a subgroup of the multiplicative group of a finite field and a character defined over . Here we provide a characterization of this aforementioned property for \textit{symmetries} arising from an index-3 subgroup and a nontrivial character .
Paper Structure (7 sections, 5 theorems, 20 equations)

This paper contains 7 sections, 5 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Preliminaries
  4. Characters and the Fourier transform
  5. Compressed Fourier transform
  6. Index-3 subgroups and nontrivial characters
  7. Acknowledgments

Key Result

Theorem 1.1

Let $p$ be a prime and $\zeta$ a primitive $p$-th root of unity. For every pair of subsets $I,J\subseteq \mathbb{F}_{p}$ with the same cardinality, the matrix $(\zeta^{ij})_{i\in I,j\in J}$ is nonsingular, that is, it has nonvanishing determinant.

Theorems & Definitions (12)

  • Theorem 1.1: Chebotarëv
  • Definition 1.2: Nonvanishing minors property
  • Definition 2.1: $\chi$-symmetry
  • Definition 2.2: Compressed Fourier transform
  • Remark 2.3
  • Proposition 2.1: GARCIA2021899
  • Lemma 3.1: GARCIA2021899
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • ...and 2 more