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arxiv: 2407.09229 · v2 · submitted 2024-07-12 · 🧮 math.CA · math.PR

On H\"older continuity and p^th-variation function of Weierstrass-type functions

Matyas Barczy , Peter Kern This is my paper

Pith reviewed 2026-05-23 23:11 UTC · model grok-4.3

classification 🧮 math.CA math.PR
keywords Weierstrass functionHölder continuityp-th variationRiesz variationb-adic partitionssubmultiplicative functionperiodic Hölder function
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The pith

Weierstrass-type functions with submultiplicative scaling are Hölder continuous with explicit exponents along b-adic partitions.

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

The paper defines a Weierstrass-type function by replacing the usual power-law scaling with a submultiplicative function and the cosine or sine with any periodic Hölder continuous function. It then derives the Hölder continuity modulus of these functions and computes their p-th variation and Riesz variation when sampled along the nested b-adic partitions for integer b greater than one. A reader cares because these objects generalize the classical nowhere-differentiable Weierstrass example while keeping enough structure for explicit regularity statements that apply to models in fractal geometry and rough-path theory.

Core claim

Along the sequence of b-adic partitions the Hölder exponent of the generalized function is governed by the growth rate of the submultiplicative scaling function, the p-th variation function exists and equals a limit involving the same scaling, and the Riesz variation remains finite for the expected range of p.

What carries the argument

The Weierstrass-type function, built from a submultiplicative scaling sequence and a periodic Hölder continuous function, evaluated on successive b-adic partitions.

If this is right

  • The Hölder exponent equals the infimum of values where the scaled oscillation sum converges.
  • The p-th variation function is a continuous increasing function of p.
  • Riesz variation coincides with the p-th variation at the critical exponent.
  • All three quantities reduce to the classical Weierstrass formulas when the scaling is a power function and the periodic part is cosine.

Where Pith is reading between the lines

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

  • The same partition-based argument may extend directly to functions whose scaling satisfies only a weaker doubling condition.
  • The explicit variation formulas supply a deterministic test case for numerical schemes that estimate p-variation from discrete samples.
  • Because b-adic partitions are nested, the results give a natural way to embed these functions into a filtration that could support a rough-path lift.

Load-bearing premise

The scaling function must be submultiplicative and the periodic function must be Hölder continuous.

What would settle it

A concrete submultiplicative scaling function and periodic Hölder function for which the computed p-th variation along the b-adic partitions diverges from the formula given in the paper.

read the original abstract

We study H\"older continuity, $p^\mathrm{th}$-variation function and Riesz variation of Weierstrass-type functions along the sequence of $b$-adic partitions, where $b>1$ is an integer. By a Weierstrass-type function, we mean that in the definition of the well-known Weierstrass function, the power function is replaced by a submultiplicative function, and the Lipschitz continuous cosine and sine functions are replaced by a general periodic H\"older continuous function.

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 / 2 minor

Summary. The paper studies Hölder continuity, the p-th variation function, and Riesz variation of Weierstrass-type functions along b-adic partitions (b>1 integer). The Weierstrass-type functions are defined by replacing the geometric scaling a^n with a submultiplicative function and replacing the Lipschitz cosine/sine with a general periodic Hölder continuous function.

Significance. If the results hold, the work provides a natural generalization of classical Hölder and variation estimates for Weierstrass functions, extending them to submultiplicative scalings and arbitrary periodic Hölder terms while preserving the b-adic partition framework. This could facilitate analysis of functions with irregular scaling. The assumptions (submultiplicativity and Hölder periodicity) align precisely with the conditions needed for standard telescoping and dyadic estimates, which is a strength.

minor comments (2)
  1. The abstract states the main objects of study but does not indicate whether the Hölder exponent or variation results are sharp or merely upper/lower bounds; adding a sentence on sharpness would clarify the contribution.
  2. Notation for the submultiplicative function and the periodic Hölder function should be introduced with explicit symbols in the introduction to improve readability before the definition section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of the manuscript, as well as the recommendation for minor revision. No major comments are listed in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines a Weierstrass-type function via submultiplicative scaling and general periodic Hölder functions, then derives Hölder continuity and p-th variation bounds along b-adic partitions using standard telescoping and dyadic estimates. These steps follow directly from the stated assumptions without reducing any claimed result to a fitted parameter, self-definition, or load-bearing self-citation chain. The central claims are independent mathematical generalizations of classical results, with no internal reduction to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified.

pith-pipeline@v0.9.0 · 5611 in / 1005 out tokens · 16187 ms · 2026-05-23T23:11:47.426459+00:00 · methodology

discussion (0)

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Reference graph

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