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Discrete Harmonic Analysis

Finite-field time-frequency independence: attempts, failure modes, and what I kept.

Background

A finite Gabor system is built from translations and modulations of one function. My project studied finite-field analogues of time-frequency linear-independence questions, where the problem becomes finite-dimensional but still retains a mix of Fourier, algebraic, and combinatorial structure.

Tools

The finite-field version of the problem forced me to get the operators straight before trying to prove anything. These are the objects that kept reappearing in my attempts.

Translation and Modulation

Definition. Let \(G\) be a finite abelian group, \(x\in G\), and \(\gamma\in\widehat G\). For \(f:G\to\mathbb C\), define \[ (T_xf)(t)=f(t-x),\qquad (M_\gamma f)(t)=\gamma(t)f(t). \]
Proposition (commutation relation). \[ M_\gamma T_x=\gamma(x)\,T_xM_\gamma. \]

Finite Gabor Systems

Definition. For nonzero \(f:G\to\mathbb C\) and \(\Lambda\subset G\times\widehat G\), \[ \mathcal G(f,\Lambda)=\{M_\gamma T_xf:(x,\gamma)\in\Lambda\}. \] The independence question asks when this family is linearly independent.

Finite Fourier Transform

Definition. With normalized counting measure, \[ \widehat f(\gamma)=\frac1{|G|}\sum_{x\in G}f(x)\overline{\gamma(x)}. \]
Theorem (finite uncertainty principle). For nonzero \(f:G\to\mathbb C\), \[ |\operatorname{supp}f|\,|\operatorname{supp}\widehat f|\geq |G|. \]

These tools made many reductions possible; the problem was that the reductions usually stopped one step before the actual independence statement.

Attempts

I spent roughly a year trying several finite-dimensional reductions of the dependence problem. The recurring failure mode was that an argument would either handle only a very structured configuration or replace the original dependence question by a rank/combinatorial condition that was nearly as hard to control.

The project produced a preliminary conference presentation and a lot of useful failed proofs. More importantly, it taught me the difference between reformulating a problem and gaining actual leverage on it.

Why I moved on

There was no single fatal counterexample. The next plausible steps increasingly looked like a pile of special cases rather than a mechanism for a general theorem, so I stopped pushing the project and moved the time into other problems.

Last updated: September 14, 2026.