Sharpness of the Cohn-Elkies linear programming bound in dimension two

Theorem 1.1 constructs a real radial Schwartz function on $\mathbb R^2$ with $\widehat f(0)=1$, $f(0)=2/\sqrt3$, $\widehat f\ge0$ and $f\le0$ outside the unit disk, so the Cohn-Elkies bound equals the triangular packing density $\pi/(2\sqrt3)$. Its zeros beyond radius one lie at integral squared radii, which recovers uniqueness of the triangular packing among periodic equality cases. It does not satisfy the stronger zero-set condition of Cohn-Elkies Conjecture 8.1, and it gives no new packing bound (the planar density was already known).

Theorem 1.1 constructs a real radial Schwartz function on $\mathbb R^2$ with $\widehat f(0)=1$, $f(0)=2/\sqrt3$, $\widehat f\ge0$ and $f\le0$ outside the unit disk, so the Cohn-Elkies bound equals the triangular packing density $\pi/(2\sqrt3)$. Its zeros beyond radius one lie at integral squared radii, which recovers uniqueness of the triangular packing among periodic equality cases. It does not satisfy the stronger zero-set condition of Cohn-Elkies Conjecture 8.1, and it gives no new packing bound (the planar density was already known).