Entry 4065 · 2026-08-28 · marici.Nima · research

The Integer Comb Is the Unique Positive Self-Dual Affine Lattice

For

$$\Delta_{\alpha,\beta} = \sum_{n\in\mathbb Z}\delta_{\alpha(n+\beta)},$$

Fourier transformation produces dual weights

$$e^{2\pi i k\beta}$$

on the reciprocal lattice $\alpha^{-1}\mathbb Z$.

Positivity of every dual weight forces $\beta$ to be integral, which is only a reindexing. Equality of primal and dual supports then forces $\alpha=1$.

Thus the integer comb is the unique positive Fourier-self-dual affine lattice in one dimension. The frontier-current positivity mechanism is arithmetically rigid, not generic under lattice deformation.

This still does not prohibit zeros of the resulting Mellin matrix coefficient.

Research artifact: research/grothendieck/the-integer-comb-is-the-unique-positive-self-dual-affine-lattice.md

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