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arxiv: 2606.08304 · v2 · pith:BE32PZFDnew · submitted 2026-06-06 · 🧮 math.CA · math.PR

Functions of Bounded Variation and Point Processes

J. Antezana , M. Levi , J. Marzo , J. Ortega-Cerd\`a This is my paper

Pith reviewed 2026-06-27 18:44 UTC · model grok-4.3

classification 🧮 math.CA math.PR
keywords bounded variationhyperuniform point processesL2-jumpBMO oscillationperimeter characterizationdifference quotientsFourier transformlinear statistics
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The pith

An averaged BMO oscillation converges to a dimensional constant times the L2-jump of a BV function when sampled against hyperuniform point processes.

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

The paper establishes characterization formulas for the jump part of the gradient of a bounded variation function, expressed both through difference quotients and through Fourier transforms. It then ties these analytic objects to hyperuniform point processes by deriving asymptotic growth rates for the variance of linear statistics, rates that are controlled by the size and location of the function's jumps. The central construction is an averaged quadratic BMO-type oscillation taken over translated and rotated cube partitions; this functional is shown to converge to an explicit constant times the L2-jump, which in turn yields a new characterization of the perimeter of a set.

Core claim

For hyperuniform point processes whose variance asymptotics are known by class, the variance of the linear statistic generated by a BV test function grows at a rate determined by the L2 norm of the jump measure of its gradient; moreover, an averaged quadratic BMO oscillation functional defined on translated and rotated cube partitions converges to an explicit dimensional constant multiplied by this same L2-jump, thereby supplying a new characterization of the perimeter of a set.

What carries the argument

The averaged quadratic BMO-type oscillation functional over translated and rotated cube partitions, whose limit recovers the L2-jump measure.

If this is right

  • The growth rate of fluctuations of linear statistics is controlled by the regularity class and the size of the jump discontinuities of the test function.
  • New explicit formulas for the L2-jump are obtained via both difference quotients and Fourier transforms, extending earlier work of Beretti-Gennaioli and Dávila.
  • The perimeter of a set admits an additional characterization as the limit of the averaged BMO oscillation functional.
  • The same limit supplies a quantitative link between the analytic jump measure and the statistical variance growth in each hyperuniformity class.

Where Pith is reading between the lines

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

  • Point-process sampling could be used to numerically approximate the total variation or perimeter of a set by evaluating the oscillation functional on finite realizations.
  • The same averaging technique might produce analogous limits for other function classes such as Sobolev or BV^p spaces.
  • The approach suggests that variance asymptotics of linear statistics could serve as a probe for jump locations even when the underlying point process is only partially observed.

Load-bearing premise

The point processes belong to one of the classified hyperuniform families whose variance asymptotics are already known, and the BV functions satisfy the integrability and approximation properties needed for the difference-quotient and Fourier expressions to equal the jump measure.

What would settle it

For the characteristic function of the unit ball, compute the averaged quadratic BMO oscillation on successively finer translated and rotated cube partitions and verify whether the values approach the predicted multiple of its L2-jump perimeter.

read the original abstract

We investigate the relationship between the analytical properties of functions of bounded variation and the statistical behavior of hyperuniform point processes. We establish several characterization formulas for the jump part of the gradient of a bounded variation function, extending and unifying previous results by Beretti--Gennaioli and D\'avila. In particular, we provide new expressions for the $L^2$-jump of the gradient using both difference quotients and Fourier transform methods. Furthermore, we connect these analytic structures to the theory of hyperuniform point processes. By analyzing the variance of linear statistics associated with bounded variation functions, we provide asymptotic estimates that depend on the specific classification of the hyperuniformity of the point process. The results show how the regularity and jump discontinuities of a function dictate the growth rate of fluctuations in point processes. Finally, we introduce an averaged quadratic BMO-type oscillation functional over translated and rotated cube partitions, similar to the one recently studied by Ambrosio et al., and prove, using results from point process, that it converges to an explicit dimensional constant times the $L^2-$jump, giving in particular a further new characterization of the perimeter of a set.

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 manuscript establishes characterization formulas for the L²-jump of the gradient of BV functions via difference quotients and Fourier methods, extending Beretti–Gennaioli and Dávila. It derives asymptotic variance estimates for linear statistics of hyperuniform point processes that depend on the hyperuniformity class and the regularity/jump structure of the test function. It defines an averaged quadratic BMO-type oscillation functional over translated and rotated cube partitions and proves, via point-process results, that the functional converges to an explicit dimensional constant times the L²-jump, yielding a new characterization of the perimeter.

Significance. If the derivations hold, the work supplies a concrete bridge between BV theory and the fluctuation theory of hyperuniform point processes, with the explicit constant in the oscillation-functional limit constituting a clear advance. The unification of difference-quotient/Fourier identities with variance asymptotics and the perimeter characterization are the primary contributions.

minor comments (3)
  1. The abstract and introduction should cite the precise theorems on hyperuniform variance asymptotics that are invoked in the linear-statistics analysis.
  2. Notation for the averaged oscillation functional (Definition in §4) would benefit from an explicit formula for the averaging measure over translations and rotations before the convergence statement.
  3. A short remark clarifying the precise integrability hypotheses on the BV function that guarantee the difference-quotient and Fourier expressions coincide with the jump measure would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were provided in the report, so we have no specific points to address point-by-point. We will incorporate any minor editorial suggestions during revision.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper extends BV jump characterizations to hyperuniform point processes by invoking external variance asymptotics for linear statistics and prior results on BMO oscillation functionals (Ambrosio et al.). No derivation step reduces by the paper's own equations to a fitted parameter or self-defined quantity; the central convergence claim to a dimensional constant times the L2-jump rests on standard point-process applications whose inputs are independent of the target result. Self-citations, if present, are not load-bearing for the main identities.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract supplies insufficient detail to list concrete free parameters or invented entities; the work rests on standard background assumptions of BV theory and hyperuniform point-process literature.

axioms (2)
  • standard math Standard decomposition and approximation properties of BV functions and their gradients
    Invoked throughout the characterizations of the jump part.
  • domain assumption Existence of hyperuniform point processes with known variance asymptotics for each classified regime
    Required for the asymptotic estimates of linear statistics.

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discussion (0)

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Reference graph

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29 extracted references · 2 canonical work pages

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