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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?

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    $\begingroup$ It is not yet known how to construct a magic function in two dimensions. As far as I know it is believed that such a function exists, based on the known existence of functions proving close-to-optimal bounds, but the method of Viazovska based on interpolation does not seem to be able to produce it as there are not enough values to interpolate, and another method has not been found. $\endgroup$
    – Will Sawin
    Commented Sep 6, 2023 at 18:01
  • $\begingroup$ @WillSawin Are there any reference about the point you mentioned? I just found that Cohn's survey article already mentioned about possibility on 2D case, but with no details. $\endgroup$
    – Seewoo Lee
    Commented Sep 6, 2023 at 18:28
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    $\begingroup$ @SamHopkins You're right, just updated :) $\endgroup$
    – Seewoo Lee
    Commented Sep 6, 2023 at 18:28
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    $\begingroup$ The conjecture that a sharp bound can be obtained from the method is already in the Cohn-Elkies paper. "Based on numerical evidence and analogy with the kissing problem, we conjecture that [the linear programming method] can also be used to get sharp bounds in dimensions 2, 8, and 24." I don't know a reference that this isn't known yet soone than the Cohn survey, but it may be possible to find one just by looking around. $\endgroup$
    – Will Sawin
    Commented Sep 6, 2023 at 18:35
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    $\begingroup$ It's actually a wide open problem in this area to find these magic functions for the $A_2$ lattice in 2 dimensions. The methods don't generalize at all. Solving this problem would be huge. $\endgroup$
    – Breakfastisready
    Commented Sep 6, 2023 at 20:05

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I may answer my own questions (after I study more about the topic). LP bound is proven to be suboptimal in dimensions

and both are based on Cohn-Triantafillou's "Dual linear programming bounds" and Cohn-de Laat-Salmon's "Three-point bounds". Dual bound also proves that LP bound cannot prove optimality of currently known best packings for $d = 12$ (Coxeter-Todd lattice), $d = 16$ (Barnes-Wall lattice), and $d = 20, 28, 32$. Also, it seems that LP bound is conjectured to be suboptimal for all the other dimensions except for $d = 1, 2, 8, 24$.

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