Some properties of optimal functions for sphere packing in dimensions 8 and 24

Abstract

We study some sequences of functions of one real variable and conjecture that they converge uniformly to functions with certain positivity and growth properties. Our conjectures imply a conjecture of Cohn and Elkies, which in turn implies the complete solution to the sphere packing problem in dimensions 8 and 24. We give numerical evidence for these conjectures as well as some arithmetic properties of the hypothetical limiting functions. The conjectures are of greatest interest in dimension 24, in light of Viazovska's recent solution to the Cohn-Elkies conjecture (and consequently the sphere packing problem) in dimension 8.

Figures (4)

• Figure 1: Number $N$ of digits to which the values of $f_k$ and $\widehat{f}_k$ agree with those of $f_{k-5}$ and $\widehat{f}_{k-5}$ at all of the points $x/10 + (y/10)i$, for integers $0 \le x \le 50$ and $0 \le y \le 2$. Data points for $n=8$ are gray and those for $n=24$ are black.
• Figure 2: The non-real roots of $f_{600}$ (above) and $\widehat{f}_{600}$ (middle) in the right half-plane, for $n=8$. The lower graph shows the points in either of the previous two graphs for which no point in the other agrees to within $10^{-6}$ in real and imaginary parts.
• Figure 3: The non-real roots of $f_{600}$ (above) and $\widehat{f}_{600}$ (middle) in the right half-plane, for $n=24$. The lower graph shows the points in either of the previous two graphs for which no point in the other agrees to within $10^{-6}$ in real and imaginary parts.
• Figure 4: The complex roots of $g^1_{8,20}$.

Theorems & Definitions (17)

• Theorem 2.1
• proof
• Conjecture 2.2: Conjecture 7.3 in CE; now a theorem when $n=8$ V
• Lemma 3.1
• proof
• Conjecture 4.1
• Conjecture 4.2
• Conjecture 4.3
• Lemma 5.1
• proof