Triangular-lattice optimality, long-range Riesz and Coulomb energies, and spherical logarithmic energy

2 · Going further

The main problem is to minimize the lower limit of energy per particle over locally finite planar configurations with centered-disk density one. The pair potential has the form $f(|x-y{|}^2)$, where $f$ is nonnegative and completely monotone: its successive derivatives alternate in sign, so $(-1{)}^k{f}^{(k)}(t)\ge 0$ for $t>0$ and every nonnegative integer $k$. The density-one triangular lattice is the minimizer stated in the main result, including comparisons in which energies are infinite. The use of a lower limit matters: the statement does not require the energy averages to converge.

The main manuscript describes an atomic interpolation certificate that constructs sharp Gaussian Fourier minorants—comparison functions providing bounds for Gaussian interactions. Manuscript 2 describes the related argument through sharp Gaussian Fourier bounds, positive mixtures, and heat-kernel comparison. It also states triangular minimality for planar logarithmic and renormalized Riesz energies, with $0<s<2$ in the Riesz case, and for the corresponding jellium minima.

The other companions address related but distinct questions. Manuscript 3 reports that the planar two-point Cohn–Elkies Fourier bound attains the circle-packing density $\pi /(2\sqrt{3})$, using a radial Schwartz certificate whose global sign conditions are established in the manuscript by rigorous interval arithmetic and analytic estimates; it also recovers classical triangular-packing uniqueness among periodic equality cases. Manuscript 4 reports the Sandier–Serfaty conclusion for a covolume-one triangular lattice among all admissible curl-free fields with a unit uniform background, using a direct Voronoi-cell comparison and rigorous interval arithmetic. Combined with Bétermin and Sandier's asymptotic formula, that manuscript obtains the Brauchart–Hardin–Saff conclusion for the linear term of optimal ordered-pair logarithmic energy on the unit two-sphere.

Useful notions

• Pair interactions and energy per point
• Density measured in expanding disks
• Limits and lower limits
• Derivatives and complete monotonicity
• Fourier bounds and Gaussian functions

What was known before / what is new

Limits and assumptions

The universal comparison requires nonnegative completely monotone functions of squared distance, local finiteness, and centered-disk density one; it is not a statement about arbitrary interaction rules or finite collections of points. It minimizes a lower limiting average and permits infinite energies. The renormalized Riesz conclusion is restricted to $0<s<2$, while the Coulomb statement concerns admissible curl-free fields with a unit uniform background. The spherical conclusion concerns the linear term in an asymptotic energy formula, not exact optimal configurations for every finite number of points. The main result has a Lean formalization, but the supplied source does not delineate its coverage of the companion claims or numerical certificates. No remaining open problem is specified, and the source does not state general uniqueness for the universal energy minimizer.

Mini glossary

Triangular lattice

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A repeating planar arrangement of points forming a grid of equilateral triangles.

Locally finite configuration

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An arrangement with only finitely many points in every bounded region.

Centered-disk density one

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The condition that the number of points inside expanding disks about a fixed center, divided by disk area, tends to one.

Lower limit

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The limiting lower value of a sequence of averages, which can be defined even when those averages do not converge.

Completely monotone function

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A smooth function whose derivatives alternate in sign, with $(-1{)}^k{f}^{(k)}(t)\ge 0$ for every nonnegative integer $k$.

Universal optimality

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Energy minimality for every potential in the specified family, under the same configuration and density conditions.

Keywords : Triangular lattice · Universal optimality · Renormalized energy

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Manuscripts in this family