# Eigenfunctions of the Fourier Transform with specified zeros

## TL;DR

This paper unifies the description of modular and quasi-modular functions used in optimal sphere packing proofs in dimensions 8 and 24, showing modular forms are essential and extending the approach to all dimensions divisible by 4.

## Contribution

It provides a unified framework for understanding the functions used in sphere packing bounds and extends these constructions to all dimensions divisible by 4.

## Key findings

- Modular forms are necessary for optimal packing bounds in dimensions 8 and 24.
- The framework can be extended to arbitrary dimensions divisible by 4.
- Unified description simplifies understanding of packing bounds proofs.

## Abstract

We give a unified description of the modular and quasi-modular functions used in Viazovska's proof of the best packing bounds in dimension 8 and the proof by Cohn, Kumar, Miller, Radchenko, and Viazovska of the best packing bound in dimension 24. We show that necessarily modular forms have to be used to obtain these results. We extend these constructions to arbitrary dimensions divisible by 4.