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arxiv: 2604.09899 · v1 · submitted 2026-04-10 · 🧮 math.CA · math.PR

The attainable almost sure large dimensions

Kathryn E. Hare , Franklin Mendivil This is my paper

Pith reviewed 2026-05-10 15:48 UTC · model grok-4.3

classification 🧮 math.CA math.PR
keywords random Moran measuresΦ-dimensionsalmost sure dimensionsAssouad dimensionMoran setsrandom fractalslarge dimensions
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The pith

The attainable almost sure large Φ-dimensions of random measures on Moran sets are completely determined, revealing a gap when weights are scale-independent.

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

The paper establishes the possible almost sure values of large Φ-dimensions for random measures constructed on random Moran sets using a natural probabilistic model. These dimensions refine local geometry by focusing on specific scale ranges, with the quasi-Assouad dimension as a key example. Results are given separately for when probability weights are tied to scaling factors and when they are chosen independently. In the independent case, a gap usually exists between the dimension of the supporting set and the minimal upper and maximal lower dimensions achievable by measures on it. The analysis also yields the almost sure dimensions of the random Moran sets as a direct consequence.

Core claim

We determine the range of possible almost sure Φ-dimensions for random measures generated by the model and supported on any given random Moran set for both dependent and independent weight cases. In the independent situation, there is usually a gap between the dimension of the set and the smallest attainable upper dimension and largest attainable lower dimension. Consequently, the almost sure dimensions of the underlying random Moran sets are determined.

What carries the argument

The natural model of random Moran measures with random scaling factors and probability weights at each construction step, together with the family of Φ-dimensions that generalize Assouad-type dimensions by selecting specific scale-depth ranges.

If this is right

  • The almost sure dimensions of the random Moran sets themselves are obtained explicitly.
  • In the case of scale-dependent weights, any value in the possible dimension interval can be attained almost surely by some measure.
  • In the independent-weights case, the attainable upper and lower dimensions leave a gap around the set dimension.
  • The results apply to any given random Moran set in the model.

Where Pith is reading between the lines

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

  • The gap in the independent case suggests that random measures cannot achieve the full range of local dimensions without correlation between weights and scales.
  • These characterizations may help analyze other random fractal measures where similar independence assumptions hold.
  • Concrete computations for specific distributions could illustrate the size of the gap in examples.

Load-bearing premise

The random Moran sets and measures are generated according to the natural model with the given distributions on scaling factors and weights, and the Φ-dimensions are well-defined on these objects.

What would settle it

A specific choice of distributions for the scaling factors and weights, followed by computation of the almost sure Φ-dimensions of the resulting measures, would show if they match or violate the predicted attainable range.

Figures

Figures reproduced from arXiv: 2604.09899 by Franklin Mendivil, Kathryn E. Hare.

Figure 1
Figure 1. Figure 1: Two examples of plots of the fj s, each with a zoom around the point, marked with a dot, of minimal dimension. and for no other j’s. When p > 1/2, p satisfies Condition j if θ ∈ [θj , θj+1) for 1 ≤ j ≤ N − 1, Condition 0 if θ < θ1 and Condition N if θ ≥ θN , and for no other j’s. Condition N holds for p = 1/2 for all θ > 0 and is the only condition which may be valid for both choices of p < 1/2 and p > 1/2… view at source ↗
read the original abstract

In this paper we study the range of possible almost sure dimensions of random measures arising from a natural model of random Moran measures. Specifically, we consider the Assouad-like ``large'' $\Phi$-dimensions of these measures. These dimensions can be tuned to consider a specific range of depths in scale and so provide refined local geometric information. The quasi-Assouad dimension is a well-known and important example of a ``large'' $\Phi$-dimension. We determine the range of possible almost sure $\Phi$-dimensions for random measures generated by the model and supported on any given random Moran set. We do this for both the case when the probability weights depend on the scaling factors and the case when they do not. In the later situation, we show that usually there is a ``gap'' between the dimension of the set and that of the smallest attainable upper dimension and largest attainable lower dimension. As a consequence of our results, we also determine the a.s. dimensions of the underlying random Moran 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 paper studies the attainable almost sure large Φ-dimensions (including quasi-Assouad-type) of random measures generated by a standard random Moran construction supported on random Moran sets. It determines explicit ranges for these dimensions in both the case of probability weights depending on the scaling factors and the independent-weights case; in the latter it establishes the existence of a gap between the almost-sure dimension of the underlying set and the smallest attainable upper Φ-dimension and largest attainable lower Φ-dimension. As a corollary the almost-sure dimensions of the random Moran sets themselves are obtained.

Significance. If the derivations hold, the work supplies precise, model-specific characterizations of refined scale-dependent dimensions for random measures and sets in a canonical probabilistic fractal setting. The explicit ranges and the gap phenomenon in the independent-weights case clarify the relationship between measure and support dimensions under different dependence structures, extending existing results on random Moran constructions. The use of standard independent choices for scalings and weights, together with almost-sure arguments, makes the claims falsifiable and directly comparable with prior literature on Φ-dimensions.

minor comments (3)
  1. [Abstract] The abstract states that a gap occurs 'usually'; the precise conditions (e.g., on the support of the distributions of scaling factors) under which the gap is guaranteed should be stated explicitly in the introduction or in the statement of the main theorem.
  2. [Introduction / Preliminaries] Notation for the Φ-dimension family and the random Moran construction (e.g., the precise form of the probability weights and the independence assumptions) should be collected in a single preliminary section with all parameters clearly labeled before the main results are stated.
  3. [Main results] The paper determines the a.s. dimensions of the random Moran sets as a consequence; a short dedicated corollary or remark summarizing these values (in terms of the given distributions) would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. Since the report lists no specific major comments, we have no individual points to address.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from model definitions

full rationale

The paper's central results determine attainable a.s. Φ-dimensions for random Moran measures via the standard probabilistic construction with independent or dependent scaling factors and weights. All claims follow directly from the given distributions, the definitions of Φ-dimensions (including quasi-Assouad), and standard almost-sure arguments on the Moran set. No equation reduces a prediction to a fitted input by construction, no uniqueness theorem is imported via self-citation, and no ansatz is smuggled in. The consequence for the underlying set's dimensions is a direct corollary, not a redefinition. This matches the expected non-circular outcome for a pure-mathematics characterization paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard properties of Moran constructions, Φ-dimension definitions, and almost-sure convergence in probability, with no free parameters or new entities introduced in the abstract.

axioms (2)
  • standard math Standard definitions and properties of Assouad-like Φ-dimensions and Moran set constructions hold as in the existing literature.
    The paper invokes these to define the random model and the dimensions under study.
  • domain assumption The random Moran sets satisfy the necessary separation and covering conditions for the dimensions to be well-defined.
    Required for the almost-sure results to apply to any given random Moran set.

pith-pipeline@v0.9.0 · 5463 in / 1378 out tokens · 47988 ms · 2026-05-10T15:48:57.813211+00:00 · methodology

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