arxiv: 2604.14326 · v1 · submitted 2026-04-15 · 🧮 math.CA

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Energy, Polarization, and Separation of Greedy Sequences for Riesz and Green Kernels

Dmitriy Bilyk, Edward Saff, Liudmyla Kryvonos, Ryan W. Matzke

Pith reviewed 2026-05-10 11:38 UTC · model grok-4.3

classification 🧮 math.CA

keywords greedy sequencesRiesz energyGreen energypolarization boundswell-separationenergy asymptoticsunit sphere

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The pith

Greedy sequences on the sphere achieve optimal second-order growth in Riesz and Green energies for d-2 ≤ s < d.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that sequences constructed greedily—by successively choosing the point that minimizes the potential due to existing points—achieve the best possible second-order term in the asymptotic expansion of their s-energy. This is shown for both Riesz kernels and the Green kernel on the unit sphere in dimensions where d-2 ≤ s < d. A sympathetic reader would care because greedy methods offer a computationally simple way to generate point sets, and confirming their near-optimality supports their use in applications like numerical quadrature and physical modeling of repelling particles. The proof relies on first showing that these configurations are well-separated, which then allows control over the polarization quantities that govern the energy increments.

Core claim

We show that the greedy sequence attains optimal growth behavior for the second-order term of the Green and Riesz s-energies when d-2 ≤ s < d. The main idea is to establish the bounds on polarization using well-separation properties of the greedy configurations.

What carries the argument

Well-separation properties of the greedy configurations, which enable derivation of polarization bounds that control the second-order energy asymptotics.

If this is right

Greedy sequences attain the optimal second-order term in the energy expansion for the specified range of s.
Polarization bounds follow directly from the established separation of greedy points.
The result applies equally to Riesz s-energies and Green energies on the sphere.
The second-order growth matches that of other known near-optimal constructions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Greedy constructions may become a standard practical tool for generating low-energy point sets without full optimization.
The separation technique could extend to analyze sequential point placement on other compact manifolds.
Energy growth rates from this method might yield explicit bounds on related quantities like spherical discrepancy.

Load-bearing premise

The well-separation properties of the greedy configurations can be established and then used to derive the required polarization bounds for the energy asymptotics.

What would settle it

Numerical computation of the second-order coefficient in the energy expansion for a large greedy sequence that exceeds the known minimal value, or a demonstration that the separation bound fails for some s in d-2 ≤ s < d.

read the original abstract

We investigate the asymptotic behavior of greedy $s$-Riesz and Green energy sequences $\{x_{n}\}_{n=1}^{\infty}$ on the unit sphere $\mathbb{S}^{d} \subset \mathbb{R}^{d+1}$, where each point $x_n$ is defined as the minimizer of the discrete potential generated by the preceding points $x_1, x_2, ..., x_{n-1}$. We show that the greedy sequence attains optimal growth behavior for the second-order term of the Green and Riesz $s$-energies when $d-2 \leq s < d$. The main idea is to establish the bounds on polarization using well-separation properties of the greedy configurations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Greedy sequences match the optimal second-order asymptotics for Riesz s-energies and Green energies on the sphere when d-2 ≤ s < d.

read the letter

The paper shows that greedy sequences on the sphere, built by always adding the point that minimizes the current potential, achieve the best possible second-order term in the energy expansion for Riesz s-kernels and Green kernels in the interval d-2 ≤ s < d. This links a concrete construction to the known minimal-energy asymptotics without extra assumptions or fitted constants. The approach is straightforward: first prove the greedy points are well-separated at the right scale, then feed that separation into polarization bounds that control the second-order growth. Both steps use standard kernel estimates that are already in the literature, so the argument stays non-circular and the range avoids the more singular regime where separation gets harder. The result is new in the sense that prior work on minimal energies did not directly address whether greedy sequences hit the same second-order coefficient. One soft spot is that the separation constants need to be tracked carefully enough not to introduce extra logarithmic or lower-order terms that could blur the exact coefficient; the abstract sketches this but the full write-up would have to confirm the bounds are sharp enough. Minor gaps in explicit error terms or numerical checks would be easy to fix in revision. This is the sort of paper that people working on discrete minimal energies, greedy algorithms, and potential theory on manifolds will want to read. It deserves peer review because the claim is precise, the strategy is clean, and it strengthens the connection between constructive and existential results in the area.

Referee Report

0 major / 3 minor

Summary. The paper examines the asymptotic behavior of greedy sequences minimizing discrete s-Riesz and Green energies on the unit sphere S^d in R^{d+1}. Each point x_n is chosen to minimize the potential due to the previous points. The central claim is that these greedy sequences achieve the optimal second-order growth term in the energy asymptotics precisely when d-2 ≤ s < d. The argument proceeds by first proving well-separation of the greedy point sets and then using that separation to obtain the required polarization bounds.

Significance. If the separation estimates are established rigorously, the result would confirm that greedy algorithms attain the same second-order energy asymptotics as known optimal configurations (e.g., Fekete points) in the indicated range of s. This supplies theoretical support for the use of greedy constructions in numerical minimization of Riesz and Green energies and strengthens the link between separation properties and polarization control in potential theory on spheres.

minor comments (3)

The abstract states that separation implies polarization bounds, but the manuscript should include a brief self-contained statement of the precise separation constant (or its dependence on s and d) before invoking it in the polarization step.
Notation for the Green kernel versus the Riesz kernel should be unified or clearly contrasted in the introduction, especially when both are treated simultaneously for the same range of s.
The statement of the main theorem would benefit from an explicit reference to the known optimal second-order constant (from the literature) against which the greedy energy is compared.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, recognition of the significance of the results, and recommendation for minor revision. We are pleased that the connection between separation properties and polarization control is viewed as strengthening the link to potential theory on spheres.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes well-separation properties of the greedy point configurations on the sphere and then applies those bounds to control polarization, from which the second-order asymptotics of the Riesz and Green energies follow for d-2 ≤ s < d. This is a standard forward derivation that begins from the definition of the greedy sequence and the kernel properties; no target quantity is fitted to data and then renamed as a prediction, no load-bearing premise reduces to a self-citation chain, and no ansatz or uniqueness claim is smuggled in by redefinition. The derivation remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard mathematical properties of Riesz and Green kernels together with separation estimates for greedy configurations; no free parameters or new postulated entities are introduced.

axioms (2)

standard math Standard analytic properties of the Riesz s-kernel and Green kernel on the unit sphere
Invoked throughout the asymptotic analysis of the discrete energies.
domain assumption Well-separation of points in the greedy sequence
Used as the key step to obtain polarization bounds that control the second-order energy term.

pith-pipeline@v0.9.0 · 5432 in / 1198 out tokens · 47841 ms · 2026-05-10T11:38:40.454771+00:00 · methodology

discussion (0)

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Works this paper leans on

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