The doubling metric and doubling measures

Arkadi Predtetchinski; J\'anos Flesch; Ville Suomala

The doubling metric and doubling measures

T0 review · 0 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Comparing sets through all doubling measures controls a new doubling metric on finite unions of open balls.

desk verdict A genuinely new doubling metric with a solid core theorem; the soft spots are minor and fixable. read the letter →

arxiv 1908.07566 v2 pith:5KRCIVJP submitted 2019-08-20 math.GN math.CAmath.MG

János Flesch , Arkadi Predtetchinski , Ville Suomala This is my paper · ORCID

classification math.GNmath.CAmath.MG MSC 54E3528A1251F99

keywords doublingmetricpredecessoroperationmeasuresimpleopensetpseudometricquasisymmetricmapporosity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces a metric on the collection of non-empty bounded open subsets of a metric space, called the doubling metric. Its directed version counts how many times one must double the radii of every open ball contained in a set before the resulting expansion contains another set. The central claim is that for finite unions of open balls — simple open sets — this geometric distance is controlled by measure comparisons: if every doubling measure gives masses to U and V that agree up to a factor $C^K$, then the doubling distance satisfies $d(U,V) \le 24K+8$. This reverses, for simple open sets, the general one-sided estimate $m \le 3d$, where $m$ records the worst-case exponent of measure comparison across all doubling measures. The paper also shows that, under mild compactness and boundedness assumptions, maps that are Lipschitz with respect to the doubling metric are exactly the continuous surjections that preserve doubling measures quantitatively.

What carries the argument

The engine is the predecessor operation $U_* = \bigcup\{O(x,2r): O(x,r)\subseteq U\}$, which expands an open set by doubling the radii of every open ball it contains; iterating it defines the directed distance $d_\to(U,V)$ and the metric $d(U,V)=\max\{d_\to(U,V),d_\to(V,U)\}$. The comparison result rests on two further objects: the measure pseudometric $m$, defined as the worst-case exponent $t$ in the inequality $\mu(U)\ge C^{-t}\mu(V)$ over all doubling measures, and the class of simple open sets, for which the closure of $U$ lies inside $U_*$ — a property that fails for general bounded open sets and explains the additive constant in the bound. The proof's second half constructs the squeezing measures $\mu_\epsilon$ by multiplying densities stepwise inside the sets $W_m$, keeping the mass of every partition element unchanged while driving down the mass of $U$; the partition geometry, encoded in conditions (P1)–(P4), is what keeps the modified measures doubling with constant $\epsilon^{-6}$.

What would settle it

Search for two simple open sets $U,V$ in a metric space with a doubling measure such that $C^{-K}\mu(U)\le \mu(V)\le C^K\mu(U)$ for every $C$-doubling measure $\mu$ yet $d(U,V)>24K+8$; any such pair refutes Theorem 3.2. A sharper probe is to test the unproved hinge directly by looking for a nonseparable metric space that carries a doubling measure.

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Extended reading notes

Core claim

The main theorem (Theorem 3.2) states that if $X$ is a metric space carrying a doubling measure and $U,V$ are simple open sets — finite unions of open balls — then $d(U,V) \le 4[6m(U,V)+2]$, where $m(U,V)$ is the infimum of all $t\ge 0$ such that $\mu(U) \ge C^{-t}\mu(V)$ for every $C\ge 1$ and every $C$-doubling measure $\mu$. In particular, if $C^{-K}\mu(U) \le \mu(V) \le C^K\mu(U)$ for all $C$ and all $C$-doubling measures, then $d(U,V) \le 24K+8$. The proof builds, for a simple open set $U$, a sequence of expanding sets $W_m$ and finite partitions of them satisfying $(W_m)_* \subseteq W_{m+1} \subseteq (W_m)^4_*$ in a controlled sense, then modifies an arbitrary doubling measure by squeezing its mass inside $W_m$ step by step. The resulting measures are $\epsilon^{-6}$-doubling for arbitrarily small $\epsilon$, while the ratio $\mu_\epsilon(U)/\mu_\epsilon(V)$ is forced to decay like $\epsilon^M$; this can only happen if $M \le 6m(U,V)$, which yields the bound.

Load-bearing premise

The proof of the key estimate $m\le 3d$ assumes, without proof, that a metric space carrying a doubling measure is separable; if that separability fact failed, the covering argument used to compare $\mu(U)$ with $\mu(U_*)$ would collapse, and with it the one-sided comparison that the main theorem reverses.

Editorial extensions

If this is right

For simple open sets, the doubling distance and the measure pseudometric are equivalent up to explicit constants: $m(U,V)\le 3d(U,V)$ always, and $d(U,V)\le 24m(U,V)+8$.
If a bound of the form $C^{-K}\mu(U)\le \mu(V)\le C^K\mu(U)$ holds for every doubling measure, then $U$ and $V$ lie within $24K+8$ doubling steps of each other.
In the presence of compact closed balls and bounded preimages of bounded sets, a continuous surjection is Lipschitz with respect to the doubling metric if and only if it preserves doubling measures quantitatively (Theorem 4.9).
Every quasisymmetric homeomorphism is bi-Lipschitz with respect to the doubling metric, via the induced map on open sets, so the doubling metric gives a quantitative bridge between measure preservation and quasisymmetry.
The doubling metric defines a porosity notion: d-porous sets are σ-upper porous, and under a divergence condition on the porosity constants they are thin for every doubling measure.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The constant $C_U=\min\{d(U,U'): U' \text{ simple}\}$ from Corollary 3.4 could be read as a quantitative 'complexity' of a general open set; if it is small, measure comparisons still nearly determine the doubling distance, suggesting a natural scale of sets between finite unions of balls and arbitrary bounded open sets.
The squeezing-measure construction is flexible: because any expansion factor larger than 1 gives a bi-Lipschitz equivalent metric (Remark 2.2(b)), the same part of the proof should yield comparability theorems with different constants for other expansion factors.
The open problem in Section 5 can be probed by seeking a uniformly perfect, non-quasisymmetric homeomorphism whose induced map is bi-Lipschitz for $d$; a positive example would show the doubling metric captures strictly more than quasisymmetry, while a proof in the ultrametric case suggests the answer may depend delicately on the geometry.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read it. The paper introduces a genuinely new metric on bounded open subsets of a metric space: for U open, U* is the union of all balls with doubled radii whose halves sit inside U; the directed distance counts iterations needed to cover V. The predecessor operation is simple, but I don't recall seeing it in the literature. The centerpiece Theorem 3.2 shows that for simple open sets (finite unions of open balls), this geometric doubling distance d and the measure-comparison pseudometric m are quantitatively equivalent: d(U,V) ≤ 4[6m(U,V)+2]. I worked through the proof; it is complete and correct. The construction of the sets W_m and partitions S_m in Part I is careful; Part II's density modification yields ε^{-6}-doubling measures, and the final contradiction gives M ≤ 6m(U,V). The stress-test note about M=0 is right: the constant K is defined as a max over an empty index set, a tiny blemish repaired by taking K=1, and the bound still holds.

The separability fact flagged in Theorem 2.6 is standard — spaces supporting doubling measures are separable — and it is not used in the proof of Theorem 3.2, so it does not threaten the main result. The proof of the one-sided bound m ≤ 3d could cite a source for separability, but this is a minor presentation gap.

What the paper does well beyond the main theorem: the game-theoretic characterization of the directed distance is a nice touch; the Lipschitz conditions (F1)–(F3) linking quantitative preservation of doubling measures to Lipschitz maps on the doubling metric are clean, and the equivalence in Theorem 4.9 gives a concrete connection to quasisymmetric maps. The examples in Section 5, especially those separating bi-Lipschitz behavior from quasisymmetry, are informative. Citations look appropriate; the self-references are to earlier work on doubling measures and porosity, and they're relevant.

Soft spots are mostly exposition: several claims in Sections 4 and 5 are asserted with 'easy to see' or 'it can be shown' rather than proved — e.g., the permutation homeomorphism check in Example 4.8 and the porosity statements in 5.2. The constants are not optimized, which the authors admit. None of this undermines the central contribution. The paper is honest about what is an open problem.

This is for analysts working with doubling measures, quasisymmetric maps, or geometric metrics on collections of sets. It deserves a serious referee, and I'd be happy to see it in print after the auxiliary claims are either proved or explicitly labeled as routine.

Referee Report

0 major / 6 minor

Summary. The paper introduces a purely metric distance between bounded open subsets of a metric space, the doubling distance, defined by iterating the predecessor operation that doubles the radii of all open balls contained in a set. It then defines a measure-theoretic variant m via the best uniform exponent governing comparability of the measures of two sets under all doubling measures. After proving the elementary estimate m(U,V) ≤ 3d(U,V), the main result (Theorem 3.2) establishes the converse comparison d(U,V) ≤ 4[6m(U,V)+2] for simple open sets, i.e., finite unions of open balls. The proof is fully written: it constructs auxiliary sets W_m and finite partitions S_m with four structural properties, then modifies a fixed doubling measure by density factors to produce ε^{-6}-doubling measures whose ratio μ(U)/μ(V) shrinks like ε^M, forcing M ≤ 6m(U,V). The final section applies the main theorem to continuous surjections, showing that under mild compactness and boundedness conditions, Lipschitzness with respect to m, preservation of doubling measures, and Lipschitzness with respect to d are equivalent, and it also defines and compares porosity notions.

Significance. The main theorem is a nontrivial and interesting quantitative bridge: it shows that for simple open sets, coarse comparability of measures under all doubling measures forces a bound on a purely combinatorial metric distance. The proof is unusually detailed and self-contained, with explicit constants throughout, and the recursive partition construction is a genuine technical contribution. If the result stands, Corollary 3.4 and Theorem 4.9 provide useful criteria for proving quantitative preservation of doubling measures from metric conditions alone. The game-theoretic reformulation of the directed distance and the porosity discussion are likely to be of independent interest.

minor comments (6)

[3.2, Part II] In the proof of Theorem 3.2, the constant K is defined as a maximum over the index set m ∈ {0,...,M−1}; if M=0 this set is empty and the subsequent choice ε < C^{-4}K^{-5} is undefined. This case can occur when V ⊆ W2 but V ⊈ W1. The gap is local: in that case d→(U,V) ≤ 8 ≤ 4[6m→(U,V)+2], so the argument should either treat M=0 separately before defining K or set K=1 for M=0 and skip the contradiction step.
[Theorem 2.6] The proof asserts without proof or reference that a metric space carrying a doubling measure is separable, and this fact is essential for the application of Lemma 2.5 to arbitrary bounded open sets. The assertion is true and standard, but a proof or citation should be supplied.
[Example 3.1] The conclusion that m(U,V)=0 when V\U is countable relies on the unstated fact that every doubling measure on the real line is non-atomic, so every countable set is thin for doubling measures. This fact is true but is not immediate from the definition of doubling measure and should be stated and justified or referenced.
[3.2, Part I] The proof of Theorem 3.2 defines M as the least natural number with V ⊆ W_{M+2} without explaining why such an M exists. Existence follows because W_m ⊇ U^m_* and d→(U,V) is finite for bounded simple sets; adding a one-sentence justification would make the proof easier to follow.
[Remark 3.3(a) and Part I] The observation that cl U ⊆ U* for simple open sets is used in the proof and is stated in Remark 3.3(a), but it is never proved. The proof is immediate for a finite union of balls and should be written out briefly.
[Various] There are a few typographical issues: in the proof of Theorem 4.9, '1/2 r' should be 'r/2', and in Example 5.1, 'coindices' should be 'coincides'.

Circularity Check

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No significant circularity: Theorem 3.2 is proved by a self-contained construction, and the few self-citations occur only in illustrative applications.

full rationale

The central claim, Theorem 3.2, is derived from the definitions of the predecessor operation, the doubling distance d, and the measure distance m, without assuming the conclusion. The proof constructs W_m, the partitions S_m, and the measures µ_ε explicitly, showing that if the measure comparability constant m(U,V) were small while d(U,V) were large, the constructed ε^{-6}-doubling measures would force a contradiction. The inequality m ≤ 3d in Theorem 2.6 is also derived from the C-doubling condition applied to the 5R-covering lemma, not from the conclusion of Theorem 3.2. The separability assertion used in Lemma 2.5 is a standard consequence of the existence of a doubling measure and is not itself imported from the paper's own prior results. Self-citations occur in Section 4 and Section 5: Example 4.4 invokes [10] for the known fact that quasisymmetric homeomorphisms satisfy (F3), and the porosity discussion cites [3] for an antecedent notion. These citations support applications and examples rather than the proof of Theorem 3.2, and they are not used to justify the main comparability result. No fitted parameter is presented as a prediction; the measure construction uses the auxiliary parameter ε only inside a contradiction argument. The derivation chain is self-contained, and any remaining technical gaps concern standard metric-space facts rather than circularity.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

No new physical or mathematical entities are postulated beyond the purely definitional doubling metric and predecessor operation, which are constructions rather than entities with independent falsifiable handles.

free parameters (1)

Radius-doubling factor in the predecessor operation = 2
Definitional choice in Section 2.1. Remark 2.2(b) notes that any factor >1 gives a bi-Lipschitz equivalent distance, so the exact value is not load-bearing.

assumptions (3)

standard math Standard measure-theoretic facts: measures are countably additive on Borel sets, doubling measures are finite on bounded balls and positive on nonempty open balls.
Used throughout; follows from definition (1.2) and standard measure theory.
standard math A metric space carrying a doubling measure is separable.
Invoked without proof in the proof of Theorem 2.6 to apply Lemma 2.5. This is a standard fact in metric measure theory.
domain assumption In Theorem 4.9, each closed ball in X is compact.
Explicit hypothesis of Theorem 4.9, used to represent preimages of closed balls by simple open sets. It is not needed for Theorem 3.2.

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Cite this review

Pith. "Pith review of The doubling metric and doubling measures." pith.science (2026). https://pith.science/paper/5KRCIVJP

@misc{pith2026190807566,
  author       = {Pith},
  title        = {Pith review of: The doubling metric and doubling measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5KRCIVJP}},
  note         = {Machine review of arXiv:1908.07566}
}

abstract

We introduce the so--called doubling metric on the collection of non--empty bounded open subsets of a metric space. Given a subset $U$ of a metric space $X$, the predecessor $U_{*}$ of $U$ is defined by doubling the radii of all open balls contained inside $U$, and taking their union. If $U$ is open, the predecessor of $U$ is an open set containing $U$. The directed doubling distance between $U$ and another subset $V$ is the number of times that the predecessor operation needs to be applied to $U$ to obtain a set that contains $V$. Finally, the doubling distance between $U$ and $V$ is the maximum of the directed distance between $U$ and $V$ and the directed distance between $V$ and $U$.

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Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

A. V . Arutyunov , S. A. Vartapetov , and S. E. Zhukovskiy . S ome properties and applications of the Hausdorff distance. J. Optim. Theory Appl., 171(2):527–535, 2016. 3This last assumption is implicit in [3, Lemma 4.1]. 21

work page 2016

[2] [2]

The role of the Pompeiu-Hausdorff metric in ﬁxed point theory

Vasile Berinde and M˘ ad˘ alina P˘ acurar. The role of the Pompeiu-Hausdorff metric in ﬁxed point theory . Creat. Math. Inform., 22(2):143–150, 2013

work page 2013

[3] [3]

On Cantor sets and doubling measures

Marianna Cs ¨ ornyei and Ville Suomala. On Cantor sets and doubling measures. J. Math. Anal. Appl. , 393(2):680–691, 2012

work page 2012

[4] [4]

Fractured fractals and broken dreams

Guy David and Stephen Semmes. Fractured fractals and broken dreams. Self-similar geometry through metric and measure, volume 7 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York, 1997

work page 1997

[5] [5]

Lectures on analysis on metric spaces

Juha Heinonen. Lectures on analysis on metric spaces . Universitext. Springer-V erlag, New York, 2001

work page 2001

[6] [6]

Two problems on doubling measures

Robert Kaufman and Jang-Mei Wu. Two problems on doubling measures. Rev. Mat. Iberoamericana, 11(3):527–545, 1995

work page 1995

[7] [7]

Lectures on quasiconformal and quasisymmetric mappings

Pekka Koskela. Lectures on quasiconformal and quasisymmetric mappings . Jyv¨ askyl¨ a Lectures in Mathematics. University of Jyv¨ askyl¨ a, Department of Mathematics and Statistics, 2009

work page 2009

[8] [8]

Every complete doubl ing metric space car- ries a doubling measure

Jouni Luukkainen and Eero Saksman. Every complete doubl ing metric space car- ries a doubling measure. Proc. Amer. Math. Soc., 126(2):531–534, 1998

work page 1998