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In a recent breakthrough, Viazovska and Cohn, Kumar, Miller, Radchenko,Viazovska solved the sphere packing problem in $\mathbb{R}^8$ and$\mathbb{R}^{24}$, respectively, by exhibiting explicit optimal functions,arising from the theory of weakly modular forms, for the Cohn-Elkies linearprogram in those dimensions. These functions have roots exactly at the lengthsof points of the corresponding optimal lattices: $\{\sqrt{2n}\}_{n\geq 1}$ forthe $E_8$ lattice, and $\{\sqrt{2n}\}_{n\geq 2}$, for the Leech lattice. Theconstructions of these optimal functions are in part motivated by the locationsof the zeros. But what are the roots of optimal functions in other dimensions?We prove a number of theorems about the location of the zeros of optimalfunctions in arbitrary dimensions. In particular, we prove that distancesbetween root lengths are bounded from above for $n \geq 1$ and not bounded frombelow for $n \geq 2$, and that the root lengths have to be arbitrarily closefor arbitrarily long, that is, for any $C, \varepsilon > 0$, there is aninterval of length $C$ in which the root lengths are at most $\varepsilon$apart. We also establish a technique that allows one to improve a non-optimalfunction in some cases.

Zeros of Optimal Functions in the Cohn-Elkies Linear Program