1.3. Strategy🔗

The proof reduces both problems to the last sign changes of Fourier eigenfunctions. Proposition 3.1 shows that a radial Schwartz function g with \widehat g = \pm g and g(0) = 0 has exponentially little L^1 mass inside B(0, c\sqrt d) whenever c < 1/\pi. For an admissible packing function F, let a = (\widehat F(0)/F(0))^{1/d} and h(x) = F(ax). Then h(0) = \widehat h(0), while the nonzero function g = \widehat h - h is anti-self-Fourier, vanishes at the origin, and is nonnegative outside B(0,1/a). Since \int g = 0, its negative part has mass \|g\|_1/2, all of which lies in this ball; the mass estimate therefore forces 1/a \ge (1/\pi - o(1))\sqrt d. In the other direction, Theorem 4.1 modifies the Mellin transform of a Gaussian to construct a Fourier pair and a self-Fourier function whose required exterior sign conditions begin at (1/\pi + o(1))\sqrt d. Stirling's formula converts the matching radius bounds into the packing exponent \sqrt{e/(2\pi)}.