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Eigenfunctions of the Fourier transform with specified zeros. (English) Zbl 1484.11108

Recent work by Cohn, Kumar, Miller, Radchenko and Viazovska has led to major breakthroughs in sphere packing, and, more generally, energy minimization problems. For instance, the optimal sphere packing arrangements have been determined in dimensions 8 and 24. The new techniques of M. S. Viazovska [Ann. Math. (2) 185, No. 3, 991–1015 (2017; Zbl 1373.52025)] among others go well beyond this; they also proved universal optimality for these arrangements.
These breakthroughs were made possible by using linear programming bounds, particularly a key result of H. Cohn and N. Elkies [Ann. Math. (2) 157, No. 2, 689–714 (2003; Zbl 1041.52011)]. In this framework, one needs to discover “magic functions” which fit the Cohn-Elkies bounds, which Viazovska et al accomplished by using novel constructions via Laplace transforms of quasimodular forms.
The current paper puts these bounds in dimensions 8 and 24, and related recent work on dimension 12, in a broader context. This leads to a construction in all dimensions which are multiples of 4. Important results concerning the positivity of Fourier coefficients of the modular forms in question are also established. These constructions are an important next step in this rapidly growing subject, and give hints of more general structure yet to be explored.

MSC:

11F03 Modular and automorphic functions
11H31 Lattice packing and covering (number-theoretic aspects)
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
11F11 Holomorphic modular forms of integral weight

Software:

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References:

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