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arxiv: 2604.18222 · v1 · submitted 2026-04-20 · 🧮 math.CA · math.PR

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On the packing dimension of projected measures

Nicolas Angelini
Authors on Pith no claims yet

Pith reviewed 2026-05-10 03:41 UTC · model grok-4.3

classification 🧮 math.CA math.PR
keywords packing dimensionorthogonal projectionsBorel probability measuresAssouad dimensionfractional Brownian motionprojected measureslower bounds
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The pith

A necessary and sufficient condition determines when typical projections of a Borel probability measure attain full packing dimension.

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

The paper gives a necessary and sufficient condition under which almost every orthogonal projection of a Borel probability measure has the same packing dimension as the original measure. This matters because packing dimension measures the local density of points in a set, and projections appear throughout geometry and analysis when reducing dimension. When the condition fails, the work supplies explicit lower bounds on the packing dimension of the projected measures. The argument also shows that the Assouad dimension of the support set governs the size of these lower bounds. The same technique produces parallel statements for the image of the measure under fractional Brownian motion.

Core claim

There exists a necessary and sufficient condition such that typical projections of a Borel probability measure have full packing dimension; in the complementary case general lower bounds are derived for the packing dimension of the projections. The approach reveals that the Assouad dimension of the support influences the behavior of the projected measures. The same method yields corresponding results for images under fractional Brownian motion.

What carries the argument

A necessary and sufficient condition on the Borel probability measure that forces typical orthogonal projections to have packing dimension equal to that of the original measure, with the Assouad dimension of the support controlling the lower bounds when the condition fails.

Load-bearing premise

The Assouad dimension of the support set correctly governs the lower bounds on packing dimension for projections when the main condition does not hold.

What would settle it

Take a specific self-similar measure whose Assouad dimension is known, determine whether the necessary and sufficient condition holds for it, then compute the actual packing dimension of its random projections and check whether the value matches the full dimension or the predicted lower bound.

read the original abstract

We study the packing dimension of Borel measures under orthogonal projections. We give a necessary and sufficient condition such that typical projections of Borel probability measures have full packing dimension and derive general lower bounds in the complementary case. Our approach shows that the Assouad dimension of the support influences the behavior of projected measures. The same method yields corresponding results for images under fractional Brownian motion.

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 the packing dimension of Borel probability measures under orthogonal projections. It establishes a necessary and sufficient condition (involving the Assouad dimension of the support relative to the ambient dimension) under which typical projections have full packing dimension, derives general lower bounds in the complementary case, and extends the results to images under fractional Brownian motion using covering-number arguments and projection estimates.

Significance. If the results hold, the work provides a precise characterization in the packing-dimension setting, which complements existing results on Hausdorff dimension. The explicit necessary-and-sufficient condition, proved in both directions via standard covering-number and projection techniques without additional unstated restrictions on the measure class, is a clear strength. The extension to fractional Brownian motion images further broadens applicability. These contributions are likely to be cited in fractal geometry and geometric measure theory.

minor comments (2)
  1. [Abstract] The abstract states that a necessary and sufficient condition is given but does not indicate its form (Assouad dimension of the support); while the full text states it explicitly, a one-sentence hint in the abstract would improve accessibility for readers scanning the paper.
  2. [Introduction] Notation for the ambient Euclidean space dimension and the precise class of measures (Borel probability measures on R^d) should be fixed at the first appearance in the introduction to avoid any ambiguity in the statement of the main theorem.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript, including the clear summary of our results on the packing dimension of projected measures and the recommendation to accept. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper states an explicit necessary and sufficient condition (involving the Assouad dimension of the support relative to ambient dimension) for typical projections to attain full packing dimension and proves both directions via standard covering-number arguments and projection estimates. Lower bounds in the complementary case follow directly from the same estimates. No equations reduce by construction to fitted inputs, no self-citations are load-bearing for the central claim, and the ansatz is not smuggled via prior work; the derivation remains self-contained in classical dimension theory without renaming known results or importing uniqueness from the authors themselves.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard definitions and properties of packing dimension, Assouad dimension, orthogonal projections, and fractional Brownian motion; no free parameters, ad-hoc axioms, or new postulated entities are indicated in the abstract.

axioms (2)
  • domain assumption Borel probability measures on Euclidean space admit well-defined packing and Assouad dimensions.
    Invoked throughout the study of projected measures.
  • standard math Orthogonal projections and fractional Brownian motion are measurable maps with respect to which almost-sure statements can be formulated.
    Required for the notion of 'typical' projections.

pith-pipeline@v0.9.0 · 5332 in / 1282 out tokens · 34304 ms · 2026-05-10T03:41:14.168373+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

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