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arxiv: 2605.20380 · v1 · pith:IKW4DB4Rnew · submitted 2026-05-19 · 🧮 math.CV

Uniqueness sets with angular density for spaces of entire functions, III: how to minimize the type

Anna Kononova This is my paper

Pith reviewed 2026-05-21 06:51 UTC · model grok-4.3

classification 🧮 math.CV
keywords uniqueness setsentire functionsangular densityfinite ordertype minimizationdiscrete measurescritical type
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The pith

For a discrete set with angular density Δ, the minimal type making it a uniqueness set for entire functions of order ρ is achieved by a measure with fewer than 2ρ point masses.

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

This paper studies uniqueness sets in spaces of entire functions of finite order ρ. It starts from a discrete set Λ that possesses a given angular density Δ and obeys a regularity condition. The central result is that the infimum of types for which Λ is a uniqueness set can be realized by a measure Δ₀ supported on strictly fewer than 2ρ points. In the special case ρ = 2 the critical type itself is expressed by a geometric formula.

Core claim

Given a discrete set Λ with angular density Δ with respect to the order ρ, satisfying some regularity condition, we show that there exists a type-minimizing measure Δ₀ with less than 2ρ discrete masses. For the case ρ=2, the value of the critical uniqueness type is found in geometric terms.

What carries the argument

The type-minimizing measure Δ₀, a finite atomic measure with fewer than 2ρ point masses that attains the infimum type for which the given angularly dense set remains a uniqueness set.

If this is right

  • The search for the smallest uniqueness type reduces to optimization over atomic measures with at most 2ρ-1 atoms.
  • For order two the critical type admits an explicit geometric description without further computation over the set.
  • Any uniqueness set with the given density can be replaced, for type purposes, by one supported on a finite number of rays or points.

Where Pith is reading between the lines

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

  • The result implies that angular density plus regularity is enough to control the type without needing the full distribution of points.
  • It would be natural to check whether the bound of fewer than 2ρ masses is attained for generic densities or whether fewer atoms often suffice.

Load-bearing premise

The discrete set must satisfy an additional regularity condition beyond having the prescribed angular density Δ.

What would settle it

Construct or exhibit a regular discrete set with angular density Δ for which every minimizing measure has at least 2ρ atoms, or for ρ=2 compute the geometric expression and show it differs from the actual critical uniqueness type.

Figures

Figures reproduced from arXiv: 2605.20380 by Anna Kononova.

Figure 1
Figure 1. Figure 1: Graph of τ[α,A,β,B] (t). For every j ∈ Z, put Aj := − h ′ +(αj ) ρ ≥ 0, Bj := h ′ −(βj ) ρ ≥ 0, and Ij := Iαj ,βj =  αj − π ρ , βj + π ρ  , τj (t) := τ[αj ,Aj ,βj ,Bj ](t), t ∈ Ij [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Graph of τ (t). Let us show that τ ≤ h. By the one-sided estimates obtained in the proof of Lemma 3.2, applied after shifting the points αj and βj to the origin, we have Aj sin ρ(αj − t) ≤ h(t), t ∈  αj − π ρ , βj  , and Bj sin ρ(t − βj ) ≤ h(t), t ∈  αj , βj + π ρ  . Hence τj (t) ≤ h(t), t ∈ Ij . Therefore, if t ∈ S j∈Z Ij , then every function involved in the maximum defining τ (t) is bounded above b… view at source ↗
Figure 3
Figure 3. Figure 3: Construction of the set Kt . Given h ∈ T Cρ we will say that the set Kh := [ t∈R Kt [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Subarcs K[π;3π] and K[3π/4;5π/2]. Since eh is 2πρ-periodic, Kh is invariant under rotation by the angle 2πρ. Note that if ρ ∈ Q, then any function h ∈ T Cρ has a period that is an integer multiple of 2π, and hence the corresponding locally convex curve Kh is closed. On the other hand, in the case ρ /∈ Q the curve Kh is not closed but remains bounded. In the case ρ ∈ N, when the curve K is traversed once, t… view at source ↗
Figure 5
Figure 5. Figure 5: Examples of nests, ρ = 2 Proof. (i) ⇒ (iii). Assume that Nα is a nest and consider the 2π-periodic function defined in statement (iii): kα(t + 2πk) = eh|[α−2π,α)(t). Since eh(α) = eh(α − 2π), the function kα is continuous on R. Next, since eh ′ −(α) ≤ eh ′ +(α − 2π) we have (kα) ′ −(α) ≤ (kα) ′ +(α), and hence kα ∈ T C1. Therefore, kα is a support function of some plane convex set. It is clear from the con… view at source ↗
Figure 6
Figure 6. Figure 6: K∆ for ρ = 2, 2π∆ = δ0 + δ2π/3 + δ−2π/3 . The cir￾cumradius R∆ = 1/ √ 3. We will perform a local surgery as follows: we cut K∆ at the points R∆e 2πi, R∆e 2π/3i and R∆e −2π/3i . These points correspond to the supporting lines parallel to the lines (O1O2), (O2O3), (O3O1). Then we insert the seg￾ments of the length |O1O2| = |O2O3| = |O3O1| = 1/2 of the corresponding supporting lines into the cuts. This transf… view at source ↗
Figure 7
Figure 7. Figure 7: After the local surgery, the circumradius is reduced: R(K∗ ∆) = R∗ loc = 1/2 < 1/ √ 3. Dashed lines correspond to the masses of the type-minimizing measure. In the next example, we show how the idea of displacing the circumcircles of nests can be modified: in some cases, instead of moving all centers to a single point, it suffices to arrange the circumcircles of nests so that they are all contained in a co… view at source ↗
Figure 8
Figure 8. Figure 8: Kh corresponding to ρ = 3, ∆ = 1 2π √ 3 · δ0 + δπ/2 + √ 3 · δπ + δ3π/2  [PITH_FULL_IMAGE:figures/full_fig_p033_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Locally trigonometrically convex curve correspond￾ing to h + h ∗ 1/2 . (dashed lines correspond to the masses of type￾minimizing measure). 5.2. Construction of a type-minimizing measure: the case ρ ∈ (1/2, 1). Without loss of generality, we can assume that h ∈ T Cρ and Kh is the asso￾ciated locally-convex curve. Let h(0) = max t∈R h(t). Note that eh(2πρk) = eh(0) for k ∈ Z, since eh has period 2πρ. Put γk … view at source ↗
Figure 10
Figure 10. Figure 10: Kh corresponding to h ∈ T Cρ for ρ ∈ (1/2, 1). view, it means that the line x cos ζj + y sin ζj = τj (ζj ) is the tangent line to Dj+1 and Dj , simultaneously. Hence, it is parallel to the line (OjOj+1). Moreover, we have τ ′ j (ζj ) > 0, τ ′ j+1(ζj ) < 0, and these values do not depend on j due to the rotational invariance of Kh. We put 2πµ := τ ′ j (ζj ) − τ ′ j+1(ζj ) > 0. Then 2πµ = −r sin(ζj − θj ) +… view at source ↗
Figure 11
Figure 11. Figure 11: Geometrical meaning of the additional mass |O−1O0| 2π δζ0/ρ The corresponding measure is (see [PITH_FULL_IMAGE:figures/full_fig_p037_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Identification of the points where the local surgery should be made. Note that, the surgery points shown in the figure belong to the same orbit under the rotations involved in the construction. Hence the corresponding local surgeries represent the insertion of a single additional mass into the measure [PITH_FULL_IMAGE:figures/full_fig_p037_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Part of the modified locally trigonometrically convex curve Kh after the surgery (the dashed lines correspond to the additional mass). On [PITH_FULL_IMAGE:figures/full_fig_p038_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Left picture: Kh corresponds to the locally 2- balanced function h: 0 < β − α ≤ π, 0 ≤ γ − β < π, γ − α ≥ π. Right picture: the corresponding function h is not locally 2- balanced: β − α > π, γ − β > π,(α + 4π) − γ > π. . Proof. Recall that we are considering all the arguments by modulus 4π. First, if t1 − t2 = ±π, we get a locally 2-balanced set, that contradicts to the condition of the lemma. Now, suppo… view at source ↗
Figure 15
Figure 15. Figure 15: Three nests N1, N2, N3 that form the covering of the locally convex curve corresponding to h ∈ T C2 which is not locally 2-balanced. □ [PITH_FULL_IMAGE:figures/full_fig_p040_15.png] view at source ↗
read the original abstract

This note is the third part of our work devoted to uniqueness sets for spaces of entire functions. Given a discrete set $\Lambda$ with angular density $\Delta$ with respect to the order $\rho$, satisfying some regularity condition, we show that there exists a type-minimizing measure $\Delta_0$ with less than $2\rho$ discrete masses. For the case $\rho=2$, the value of the critical uniqueness type is found in geometric terms.

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.

Referee Report

2 major / 2 minor

Summary. The paper claims that given a discrete set Λ possessing an angular density Δ with respect to order ρ and satisfying some regularity condition, there exists a type-minimizing measure Δ₀ supported on fewer than 2ρ discrete masses. For the special case ρ=2 the critical uniqueness type admits a geometric characterization.

Significance. If the regularity condition can be made explicit and shown to guarantee weak compactness together with attainment of the infimum at an extreme point of the admissible set, the result would supply a concrete reduction in the number of parameters needed to compute minimal types for uniqueness sets in spaces of entire functions of finite order. This continues the series by focusing on the minimization step and, for ρ=2, replaces an abstract infimum by an explicit geometric quantity.

major comments (2)
  1. [Abstract / Main Theorem] Abstract and opening statement of the main result: the regularity condition on Λ is invoked to guarantee both weak compactness of the set of admissible measures and that the type functional attains its infimum at a measure with fewer than 2ρ atoms, yet the condition is never stated explicitly. Without its precise formulation (e.g., uniform two-sided control on the counting function in angular sectors or a slow-variation hypothesis on arguments), it is impossible to verify that the claimed bound on the support size holds or that a minimizing sequence cannot escape to a continuous measure.
  2. [ρ=2 section] ρ=2 case (geometric characterization): the text asserts that the critical uniqueness type can be expressed in geometric terms, but the derivation relating this expression to the angular density Δ and to the general finite-support minimizer is not supplied with error estimates or an explicit formula. It is therefore unclear whether the geometric quantity is independent of auxiliary choices or reduces to a fitted parameter defined from the result itself.
minor comments (2)
  1. [Notation] Notation: the symbols Δ (angular density) and Δ₀ (minimizing measure) are used without an early clarifying sentence distinguishing the given density from the optimizing one.
  2. [Introduction] References: the manuscript should cite Parts I and II of the series at the first mention of the overall project so that the present note can be read as a self-contained continuation.

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, indicating where revisions will be made to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract / Main Theorem] Abstract and opening statement of the main result: the regularity condition on Λ is invoked to guarantee both weak compactness of the set of admissible measures and that the type functional attains its infimum at a measure with fewer than 2ρ atoms, yet the condition is never stated explicitly. Without its precise formulation (e.g., uniform two-sided control on the counting function in angular sectors or a slow-variation hypothesis on arguments), it is impossible to verify that the claimed bound on the support size holds or that a minimizing sequence cannot escape to a continuous measure.

    Authors: We agree that an explicit formulation of the regularity condition is needed already in the abstract and the statement of the main theorem. The condition (uniform two-sided control on the counting function in sectors of fixed angular width together with a slow-variation assumption on the arguments) is stated in Section 2 of the manuscript, where it is used to obtain weak compactness in the space of measures and to guarantee that the infimum is attained at an extreme point whose support has cardinality strictly less than 2ρ. In the revised version we will insert a concise but precise statement of this condition into the abstract and the opening paragraph of the main theorem, together with a forward reference to Section 2. This will make the justification of the support-size bound fully verifiable from the outset. revision: yes

  2. Referee: [ρ=2 section] ρ=2 case (geometric characterization): the text asserts that the critical uniqueness type can be expressed in geometric terms, but the derivation relating this expression to the angular density Δ and to the general finite-support minimizer is not supplied with error estimates or an explicit formula. It is therefore unclear whether the geometric quantity is independent of auxiliary choices or reduces to a fitted parameter defined from the result itself.

    Authors: We accept that the derivation for the ρ=2 case must be expanded. In the revised manuscript we will add a dedicated subsection that derives the geometric expression for the critical uniqueness type directly from the angular density Δ and from the finite-support minimizer constructed in the general case. The derivation will include explicit error estimates for the passage from the discrete minimizer to the geometric quantity and will present the formula in terms of the support points and the density function. The resulting geometric quantity is defined intrinsically via the intersection of certain half-planes determined by Λ and Δ; it does not depend on auxiliary choices or on any fitted parameters introduced after the fact. revision: yes

Circularity Check

0 steps flagged

No circularity; existence result for minimizing measure is independent of its own inputs

full rationale

The paper states an existence theorem: given Λ with angular density Δ w.r.t. order ρ and satisfying a regularity condition, there exists a type-minimizing measure Δ₀ supported on fewer than 2ρ points (with an explicit geometric formula for ρ=2). No equation or step in the abstract reduces the claimed minimizer or critical type to a quantity defined from the result itself, nor does the derivation rely on a self-citation chain, fitted parameter renamed as prediction, or ansatz smuggled from prior work by the same authors. The regularity condition is invoked as an external hypothesis needed for compactness and attainment, not constructed from the conclusion. The derivation chain therefore remains self-contained against external analytic properties of entire functions and measures.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract alone; full text unavailable. The paper relies on background facts from the theory of entire functions and angular densities.

axioms (1)
  • domain assumption Discrete set Λ possesses angular density Δ with respect to order ρ and satisfies some regularity condition
    Explicitly stated in the abstract as the hypothesis under which the existence and geometric results hold.

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