Viazovska constructed magic functions via integral transforms of (quasi-)modular forms that gives a tight bound for linear programming bounds in 8 and 24 dimensions, which resolved sphere packing problems in these dimensions and awarded Fields medal. I found that there are many ongoing works in these direction, e.g. for other dimensions, or more general theory like Fourier interpolation.

I wonder if we can re-prove results for sphere packing in low-dimensions, especially 2 (by Thue) and 3 (by Hales). Since it is conjectured that for dim 3 case (and any other higher dimensions $\neq$ 8, 24) linear programming bounds are suboptimal, we may only consider 2-dimensional case. Would it be still possible to construction "magic function" in dimension 2?

Viazovska constructed magic functions via integral transforms of (quasi-)modular forms that gives a tight bound for linear programming bounds in 8 and 24 dimensions (with other mathematicians after two weeks), which resolved sphere packing problems in these dimensions and awarded Fields medal. I found that there are many ongoing works in these direction, e.g. for other dimensions, or more general theory like Fourier interpolation.

I wonder if we can re-prove results for sphere packing in low-dimensions, especially 2 (by Thue) and 3 (by Hales). Since it is conjectured that for dim 3 case (and any other higher dimensions $\neq$ 8, 24) linear programming bounds are suboptimal, we may only consider 2-dimensional case. Would it be still possible to construct "magic function" in dimension 2?