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arxiv: 2604.12997 · v1 · submitted 2026-04-14 · 🧮 math.CA · math.AP

Uniqueness and non-uniqueness pairs for the fractional Laplacian

Ricardo Motta This is my paper

Pith reviewed 2026-05-10 13:37 UTC · model grok-4.3

classification 🧮 math.CA math.AP
keywords fractional Laplacianuniqueness pairsnon-uniquenessdiscrete subsetsmultiplier operatorsR^dvanishing conditions
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The pith

Discrete sets Lambda and M in R^d can be chosen so that any function vanishing on Lambda with its fractional Laplacian vanishing on M must be identically zero, or counterexamples can be built showing non-uniqueness.

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

The paper establishes sufficient conditions on two discrete subsets Lambda and M of R^d such that any function f vanishing on Lambda and with its fractional Laplacian vanishing on M must be zero everywhere. This characterizes uniqueness for the fractional Laplacian via discrete data. It also supplies explicit examples of discrete sets where non-trivial functions satisfy the same vanishing conditions, demonstrating non-uniqueness. The proof ideas extend to a wider class of multiplier operators.

Core claim

Assuming f=0 on Lambda and (-Delta)^s f=0 on M, where Lambda and M are discrete subsets of R^d, sufficient conditions on these sets force f to vanish identically, while other choices of the sets permit non-zero functions satisfying the same conditions. Some of the ideas used in the proofs also extend to a broader class of multiplier operators.

What carries the argument

Uniqueness or non-uniqueness pairs of discrete sets (Lambda, M) for the fractional Laplacian, where the pair determines whether the vanishing of f on Lambda together with the vanishing of (-Delta)^s f on M implies f is identically zero.

Load-bearing premise

The discrete sets Lambda and M must satisfy specific distribution conditions in R^d that are sufficient to prevent non-trivial functions from satisfying both vanishing conditions at once.

What would settle it

A non-zero function f that is zero on Lambda and whose fractional Laplacian is zero on M, for any pair of sets the paper claims forms a uniqueness pair.

read the original abstract

We establish sufficient conditions on discrete subsets of $\mathbb{R}^d$ for them to form a uniqueness or a non-uniqueness pair for the fractional Laplacian. Specifically, assuming that $f=0$ on $\Lambda$ and that $(-\Delta)^sf=0$ on $M$, where $\Lambda, M \subset \mathbb{R}^d$ are discrete, we find sufficient conditions on these sets that force $f$ to vanish identically, and we provide examples in which non-uniqueness occurs. Some of the ideas used in the proofs also extend to a broader class of multiplier operators.

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

0 major / 3 minor

Summary. The paper establishes sufficient conditions on discrete subsets Λ, M ⊂ ℝ^d such that if a function f vanishes on Λ and its fractional Laplacian (−Δ)^s f vanishes on M, then f ≡ 0 (uniqueness pair). It also supplies explicit examples of non-uniqueness pairs and notes that some techniques extend to a broader class of multiplier operators.

Significance. If the stated conditions and examples hold, the work contributes concrete criteria for unique continuation and non-uniqueness phenomena for the fractional Laplacian sampled on discrete sets. The combination of positive uniqueness results with explicit counterexamples for non-uniqueness is useful for delineating the boundary between the two regimes, and the extension to multiplier operators increases the scope beyond the fractional Laplacian alone.

minor comments (3)
  1. §1: The precise statement of the growth or density conditions on Λ and M (e.g., separation or Beurling-type density) should be recalled explicitly in the introduction rather than deferred entirely to the statements of the theorems.
  2. §4, Example 4.2: The construction of the non-uniqueness pair would benefit from a brief remark on whether the same example works for all s ∈ (0,1) or only for a restricted range of s.
  3. Notation: The symbol for the fractional Laplacian is introduced without an explicit integral or Fourier definition in the preliminary section; adding a short display equation would improve readability for readers outside the immediate area.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of its contributions to uniqueness and non-uniqueness pairs for the fractional Laplacian on discrete sets, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives sufficient conditions on discrete sets Lambda and M such that vanishing of f on Lambda together with vanishing of (-Delta)^s f on M forces f identically zero (or provides counterexamples for non-uniqueness). These conditions are obtained via analysis of the fractional Laplacian and extend to multiplier operators. No load-bearing step reduces by the paper's own equations to a definition, a fitted input renamed as prediction, or a self-citation chain. The result is framed as existence of analytic conditions rather than a tautological or uniqueness-imported claim, making the derivation independent of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; all details are deferred to the full manuscript.

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discussion (0)

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