Math Conjectures

High-Dimensional Sphere-Packing Exponent

OPENLandmarkOpen problemProposed c. 1900 · Standard version

Canonical statement

Let Δd\Delta_d be the supremal density of a packing of congruent balls in Rd\mathbb R^d. Determine the exponential asymptotics of Δd\Delta_d, equivalently determine
lim inf⁡d→∞1dlog⁡2Δdandlim sup⁡d→∞1dlog⁡2Δd, \liminf_{d\to\infty}\frac1d\log_2\Delta_d\quad\text{and}\quad\limsup_{d\to\infty}\frac1d\log_2\Delta_d,
and decide whether these two quantities are equal.
View source LaTeX
Let \(\Delta_d\) be the supremal density of a packing of congruent balls in \(\mathbb R^d\). Determine the exponential asymptotics of \(\Delta_d\), equivalently determine
\[
  \liminf_{d\to\infty}\frac1d\log_2\Delta_d\quad\text{and}\quad\limsup_{d\to\infty}\frac1d\log_2\Delta_d,
\]
and decide whether these two quantities are equal.
DG-018:[CohnElkies2003Bounds][KabatianskyLevenshtein1978Bounds][OpenAI2026TenAdvances]
Discrete geometryInformation theoryAnalysis and functional analysis

Notes

The fixed-dimensional sphere-packing catalog entry DG−013DG-013 asks for the optimum in dimension five; this entry instead asks for the true exponential density rate as dimension grows. Cohn and Elkies introduced the Fourier linear-programming upper bound [CohnElkies2003Bounds], building on the asymptotic regime of Kabatiansky and Levenshtein [KabatianskyLevenshtein1978Bounds]. The August 1, 2026 manuscript claims a better upper exponent and an exact limit for that method [OpenAI2026TenAdvances], but neither determines the true packing exponent.

Proof-claim watch (1)

High-dimensional Cohn–Elkies linear-programming exponent

Grade C

Substantive same-day claim; even if correct, it improves an upper bound and a method limit rather than determining the true packing exponent.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.