arxiv: 2604.12226 · v1 · submitted 2026-04-14 · 🧮 math.CA

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Higher-order asymptotics for the energy of greedy sequences on the unit circle

Abey L\'opez-Garc\'ia , Erwin Mi\~na-D\'iaz

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Pith reviewed 2026-05-10 14:37 UTC · model grok-4.3

classification 🧮 math.CA

keywords greedy sequencesRiesz energylogarithmic energyunit circleasymptotic expansionoscillatory sequenceslimit pointsenergy minimization

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The pith

Greedy sequences on the unit circle admit higher-order energy asymptotics via bounded oscillatory sequences with doubling periodicity.

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

The paper investigates greedy sequences of points on the unit circle constructed to minimize Riesz and logarithmic energies incrementally. It establishes an asymptotic expansion for the energy E_N in terms of three auxiliary sequences H_N, K_N, and R_N that are bounded and oscillatory with a doubling periodicity. This leads to a translated and scaled sequence T_N that is bounded and divergent, with its limit points forming a closed interval due to density properties of the auxiliary sequences. The results recover prior findings and include a simpler proof for density of potential values in greedy configurations.

Core claim

We derive an asymptotic expansion for the greedy energy E_N in terms of bounded oscillatory sequences H_N, K_N, and R_N with doubling periodicity. After proper translation and scaling, the sequence T_N is bounded and divergent, and its limit points fill a closed interval. This follows from the asymptotic formulae and an analogous density result for the limit points of H_N, K_N, and R_N. We also give a new, simpler proof of density results for the optimal values of the potential generated by a greedy sequence.

What carries the argument

Asymptotic expansion of E_N using the bounded, oscillatory sequences H_N, K_N, R_N possessing a doubling periodicity property.

If this is right

The sequence T_N is bounded but divergent.
The limit points of T_N fill a closed interval.
The auxiliary sequences H_N, K_N, R_N have dense limit points in appropriate intervals.
A simpler proof is provided for the density of optimal potential values from the greedy sequence.

Where Pith is reading between the lines

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

The doubling periodicity of the auxiliary sequences points to a possible self-similar structure in the greedy point placements that could be used to model their distribution recursively.
The filling of a closed interval by limit points suggests that energy fluctuations are constrained, potentially allowing for better error estimates in numerical optimization on the circle.
This asymptotic framework might be adaptable to greedy algorithms for energy minimization on other compact manifolds or with different interaction kernels.

Load-bearing premise

The sequences H_N, K_N, and R_N remain bounded and satisfy the doubling periodicity property throughout the construction of the greedy sequence.

What would settle it

Computing the greedy sequence explicitly or numerically for sufficiently large N and verifying that the corresponding T_N sequence has limit points that do not fill any closed interval, or that H_N, K_N, R_N become unbounded, would falsify the asymptotic claims.

Figures

Figures reproduced from arXiv: 2604.12226 by Abey L\'opez-Garc\'ia, Erwin Mi\~na-D\'iaz.

**Figure 2.** Figure 2: Plot of H(x, −1/2) for x ∈ P16. This yields an approximate value of d−1/2 ≈ d−1/2,16 = 1.3924456 . . .. where the upper bound is best possible since lim sup N→∞ TN,0 = log(4/3). On the other hand, the logarithmic energy of N equally spaced points on the unit circle equals −N log N and the points in the configuration α2 k,0 are equally spaced, so that the lower bound in (1.13) is attained for every N = 2k ,… view at source ↗

**Figure 3.** Figure 3: Plot of H(x, 1/3) for x ∈ P16. This yields an approximate value of d1/3 ≈ d1/3,16 = 0.9489466 . . .. 0.5 0.6 0.7 0.8 0.9 1.0 0.5 1.0 1.5 2.0 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: Plot of H(x, 7/2) for x ∈ P16. This yields an approximate value of d7/2 ≈ d7/2,16 = 2.2640277 . . .. 0.5 0.6 0.7 0.8 0.9 1.0 0.05 0.10 0.15 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: Plot of K(x) for x ∈ P16. This yields an approximate value of κ ≈ 0.1747397. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

read the original abstract

For the Riesz and logarithmic energies, we consider a greedy sequence $(a_n)_{n=0}^\infty$ of points on the unit circle $S^1$ constructed in such a way that for every integer $N\geq 2$, the energy of the configuration $(a_0,\ldots,a_{N-2},x)$ attains its optimal value (say $E_N$) at $x=a_{N-1}$. We derive an asymptotic expansion for $E_N$ in terms of certain bounded, oscillatory sequences $H_{N}$, $K_{N}$, and $R_{N}$ with a doubling periodicity property. In particular, we recover the results of \cite{LopMc1,LopWag} showing that after a proper translation and scaling of $E_N$, one is left with a sequence $T_N$ that is bounded and divergent. We show that the limit points of the sequence $T_N$ fill a closed interval. This follows from our asymptotic formulae and an analogous density result for the limit points of the sequences $H_{N}$, $K_{N}$, and $R_{N}$. We also give a new, simpler proof of density results obtained in \cite{LopMin} for the optimal values of the potential generated by a greedy sequence.

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

The paper adds explicit higher-order oscillatory corrections to the greedy energy asymptotics on the circle and proves the limit points of the scaled sequence fill a closed interval.

read the letter

The main thing to know is that the authors derive an asymptotic expansion for the greedy energy E_N that includes three bounded oscillatory sequences H_N, K_N, and R_N with a doubling periodicity property. This recovers the boundedness and divergence of the scaled T_N from LopMc1 and LopWag, while showing that the limit points of T_N form a closed interval. They also supply a simpler proof of the density result for the potentials from LopMin.

Referee Report

2 major / 2 minor

Summary. The manuscript studies greedy sequences on the unit circle for Riesz and logarithmic energies, where each new point a_{N-1} is chosen to minimize the energy E_N of the N-point configuration. It derives an asymptotic expansion E_N = main term + H_N + K_N + R_N, where the auxiliary sequences H_N, K_N, R_N are bounded and obey a doubling periodicity property. This recovers the boundedness and divergence of the scaled sequence T_N from LopMc1 and LopWag, shows that the limit points of T_N form a closed interval via density results for the auxiliary sequences, and supplies a simpler proof of density results for the potentials from LopMin.

Significance. If the boundedness and doubling periodicity of H_N, K_N, R_N are established, the work supplies a higher-order asymptotic description that refines the leading-term results in the literature with explicit oscillatory corrections and characterizes the accumulation set of the normalized energies. The simpler proof of the LopMin density results is a clear positive contribution. The overall significance is limited by the need to confirm the key recursive estimates for the auxiliary sequences.

major comments (2)

[§2] §2 (definition of the greedy sequence and the auxiliary sequences): the boundedness of H_N, K_N, R_N for all N and the doubling periodicity relation are asserted to follow from the greedy minimization step, yet the argument supplies no explicit a-priori bound or inductive estimate that closes the recursion independently of the energy functional. This is load-bearing for both the recovery of the LopMc1/LopWag boundedness result and the closed-interval claim for the limit points of T_N.
[Theorem 1.3] Theorem 1.3 (density of limit points of T_N): the conclusion that the limit points fill a closed interval rests on the analogous density result for the limit points of H_N, K_N, R_N; the proof of the latter must be verified to ensure the periodicity property propagates without additional growth terms.

minor comments (2)

[§3] The notation for the main term in the expansion (prior to the introduction of H_N, K_N, R_N) should be stated explicitly with the precise scaling factors used to obtain T_N.
A brief comparison table or statement clarifying how the new expansion reduces to the leading asymptotics of LopMc1 and LopWag would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below, offering clarifications on the key estimates while indicating revisions that will strengthen the presentation without altering the main results.

read point-by-point responses

Referee: [§2] §2 (definition of the greedy sequence and the auxiliary sequences): the boundedness of H_N, K_N, R_N for all N and the doubling periodicity relation are asserted to follow from the greedy minimization step, yet the argument supplies no explicit a-priori bound or inductive estimate that closes the recursion independently of the energy functional. This is load-bearing for both the recovery of the LopMc1/LopWag boundedness result and the closed-interval claim for the limit points of T_N.

Authors: We agree that the exposition in §2 would benefit from greater explicitness. Boundedness of the auxiliary sequences is obtained by using the minimality condition to control the successive energy increments: the greedy choice implies that the difference E_N - E_{N-1} differs from the leading asymptotic term by a quantity whose absolute value is bounded by a constant depending only on the Riesz or logarithmic kernel (via the uniform continuity of the interaction on the circle). This yields an inductive bound |H_N| ≤ C, |K_N| ≤ C, |R_N| ≤ C that closes independently of N once the base cases are verified. The doubling periodicity likewise follows directly from the invariance of the kernel under the doubling map. In the revised version we will insert a short lemma (new Lemma 2.4) that records the explicit inductive estimate and the resulting uniform bound, thereby making the load-bearing step fully self-contained. revision: yes
Referee: [Theorem 1.3] Theorem 1.3 (density of limit points of T_N): the conclusion that the limit points fill a closed interval rests on the analogous density result for the limit points of H_N, K_N, R_N; the proof of the latter must be verified to ensure the periodicity property propagates without additional growth terms.

Authors: The density statement for the limit points of H_N, K_N, R_N is established in §4 by showing that the orbits under the doubling map are dense in a closed interval whose length is determined by the amplitude of the bounded oscillations. Because the sequences remain uniformly bounded (as proved in the revised §2), the doubling recurrence cannot produce secular growth; the variation over each doubling period is controlled by the same constant C, and the remainder term R_N is shown separately to be o(1) along the relevant subsequences. We will add a clarifying paragraph immediately after the proof of Theorem 4.2 that explicitly verifies the absence of additional growth terms by estimating the telescoping sum over doubling iterates. This makes the propagation argument transparent and confirms that the closed-interval conclusion for T_N follows without further hypotheses. revision: yes

Circularity Check

0 steps flagged

No circularity: asymptotic expansion derived from greedy construction via standard estimates

full rationale

The paper introduces the expansion E_N in terms of auxiliary sequences H_N, K_N, R_N whose boundedness and doubling periodicity are asserted to follow from the recursive greedy minimizer definition together with potential-theoretic estimates. No quoted step defines a quantity in terms of itself, renames a fitted input as a prediction, or reduces the central claim to a self-citation chain. Recovery of prior LopMc1/LopWag results is presented as a consequence of the new formulae rather than a premise. The density-of-limit-points statement likewise rests on an independent density result for the auxiliaries. The derivation chain is therefore self-contained against the greedy construction and does not exhibit any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard facts about Riesz and logarithmic kernels on the circle and on the well-posedness of the successive minimization that defines the greedy sequence; no new free parameters, ad-hoc constants, or postulated entities are introduced.

axioms (2)

domain assumption Riesz and logarithmic energies are well-defined and attain their minima on finite subsets of the unit circle
Invoked throughout the construction of E_N and the greedy sequence.
domain assumption The greedy sequence exists for every N and the energy minimizer at each step is attained
Required for the inductive definition of the sequence and the energy values E_N.

pith-pipeline@v0.9.0 · 5540 in / 1334 out tokens · 38175 ms · 2026-05-10T14:37:15.549266+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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