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Both finite supports cannot be singletons when $N > 1$

Statement refuted

For every $N \geq 1$ there exists a nonzero $f \in C^{Z / N Z}$ with $∣ supp f ∣ = ∣ supp F_{N} f ∣ = 1$ .

Facts & Assumptions

Given: An integer $N \geq 1$ and the unitary discrete Fourier transform $F_{N}$ of The unitary discrete Fourier transform on $Z / N Z$ (The congruence class $[a]_{n}$ and the quotient set $Z / n$ ).

[F1]

For every nonzero $f \in C^{Z / N Z}$ the support product satisfies $∣ supp f ∣ \cdot ∣ supp F_{N} f ∣ \geq N$ (Finite support-product uncertainty for the unitary DFT).

[F2]

At $N = 1$ there is exactly one class, and the delta $δ_{0}$ at it is the constant function $1$ ; the example Delta and constant functions are finite DFT extremisers computes $F_{1} δ_{0} = δ_{0}$ , so $∣ supp δ_{0} ∣ = ∣ supp F_{1} δ_{0} ∣ = 1$ .

Counterexample

technique · direct

1.1F1given

Impossibility for $N > 1$ . Suppose $N > 1$ and a nonzero $f$ satisfied $∣ supp f ∣ = ∣ supp F_{N} f ∣ = 1$ . Then the left-hand side of the bound [F1] equals $1$ , so $1 \geq N$ , contradicting $N > 1$ . Hence no such $f$ exists for $N > 1$ .

1.2F2given

The case $N = 1$ . At $N = 1$ the group $Z / 1 Z$ has the single class $[0]$ , and by [F2] the delta $δ_{0}$ is the constant function $1$ with $F_{1} δ_{0} = δ_{0}$ ; both its support and the support of its transform equal the one-element set ${[0]}$ .

2.1step 1.1step 1.2∎

Conclusion. The universal claim fails already at $N = 2$ , where [F1] forces a support product of at least $2$ ; step 1.2 shows that the hypothesis $N > 1$ is essential, since the excluded configuration does occur at $N = 1$ and the bound of [F1] is exactly attained there.

Depends on

Used by

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Dependency tree · two levels

26 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Terence Tao, An Uncertainty Principle for Cyclic Groups of Prime Order, Math. Res. Lett. 12 (2005) 121–127 (arXiv:math/0308286) (standard reference, not scraped)

Both finite supports cannot be singletons when N>1N>1