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Measure doubling in unimodular locally compact groups and quotients

T0 review · 2 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A compact symmetric set with doubling $K$ projects to quotient groups with doubling at most $K^2$, and this constant is optimal.

desk verdict Sharpness construction has a fixable gap, but the main quotient doubling bounds are new, correct, and worth refereeing. read the letter →

arxiv 2411.17246 v1 pith:43AV5TVC submitted 2024-11-26 math.GR math.COmath.LO

Zuxiang Kong , Fei Peng , Chieu-Minh Tran This is my paper · ORCID
classification math.GRmath.COmath.LO MSC 22D0551F9922E3003C9911B30
keywords measuredoublinglocallycompactgroupsHaarquotientRuzsadistancespillovertechniquesmallunimodular
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

This paper establishes a sharp quotient version of small-doubling for sets in unimodular locally compact groups, i.e., groups whose Haar measure is both left- and right-invariant. If a compact set $A$ satisfies $A=A^{-1}$ and $\mu_G(A^2)\le K\mu_G(A)$, then for every closed normal subgroup $H$ the projected set satisfies $\mu_{G/H}((\pi A)^2)\le K^2\mu_{G/H}(\pi A)$, and this $K^2$ cannot be improved to $(1-\varepsilon)K^2$ for any $\varepsilon>0$. Without symmetry the bound becomes $K^3$, improving a previous $32K^6$ bound from connected noncompact Lie groups and covering all unimodular locally compact groups, including discrete ones. A further theorem extracts a subset $B\subseteq A$ of more than half the measure of $A$ whose projection has doubling less than $2K$. These results quantify exactly how small-doubling sets behave under quotient maps, tightening a tool used in the structural classification of approximate groups.

What carries the argument

The engine of the proof is the fiber-length slicing of sets over the quotient. For $gH\in G/H$, the fiber length $f_A(gH)=\mu_H(g^{-1}A\cap H)$ counts how much of $A$ lies above the coset, and Fubini's theorem gives $\mu_G(A)=\int_0^\infty \mu_{G/H}(\{f_A\ge t\})\,dt$. The Spillover Inequality (Lemma 3.3) bounds $\mu_G(AB)$ below by the integral over $t$ of $\mu_{G/H}(\pi A\cdot\{f_B\ge t\})$, connecting the product structure in $G$ to products of level sets in the quotient. The proof picks a level set for which a product-measure estimate holds, then applies the Ruzsa triangle inequality to compare $\pi A$ with its inverse, yielding the $K^2$ (or $K_1K_2$) bound. Measurability of products is handled by $\sigma$-compact approximations of the superlevel sets obtained from inner regularity; a set-theoretic remark shows the theorem is absolute across models of set theory.

What would settle it

Compute $\mu_G(A^2)$ for the Proposition 2.1 construction with $H$ a nontrivial finite group (for instance $\mathbb{Z}/2\mathbb{Z}$): if the result is strictly larger than $K\mu_G(A)$, then the example does not satisfy the hypothesis $\mu_G(A^2)=K\mu_G(A)$, showing the sharpness claim as written depends on $H$ being atomless. For the upper bound, any symmetric compact $A$ with $\mu_G(A^2)\le K\mu_G(A)$ whose projection satisfies $\mu_{G/H}((\pi A)^2)>K^2\mu_{G/H}(\pi A)$ would refute Theorem 1.1.

Watch

Extended reading notes

Core claim

At the core is Theorem 1.1: for a unimodular locally compact group $G$ with Haar measure $\mu_G$, a compact symmetric set $A$ with $\mu_G(A^2)\le K\mu_G(A)$, and a closed normal subgroup $H$ with quotient map $\pi$, one has $\mu_{G/H}((\pi A)^2)\le K^2\mu_{G/H}(\pi A)$, and $K^2$ is optimal as a universal constant. The same machinery gives the mixed bound $\mu_{G/H}((\pi A)^2)\le K_1K_2\mu_{G/H}(\pi A)$ when $\mu_G(A^2)\le K_1\mu_G(A)$ and $\mu_G(A^{-1}A)\le K_2\mu_G(A)$, from which the $K^3$ bound follows by a Ruzsa-distance estimate on $A^{-1}A$. The paper also proves that from any compact $A$ with doubling $K$ one can extract a compact $B\subseteq A$ with $\mu_G(B)>\mu_G(A)/2$ and $\mu_{G/H}((\pi B)^2)<2K\mu_{G/H}(\pi B)$. The sharpness example realizes quotient doubling $K^2-2K+2$, asymptotic to $K^2$.

Load-bearing premise

The main theorem requires the group's Haar measure to be the same from the left and from the right (unimodularity), because the proof uses that symmetry in the key inequality; the sharpness example also needs the compact subgroup $H$ to have no atoms, since the calculation treats the single point $\{1_H\}$ as having zero measure.

Editorial extensions

If this is right

  • Symmetric small-doubling is preserved under quotient maps with a universal loss of $K^2$: any compact symmetric $A$ with doubling $K$ has image with doubling at most $K^2$ in every quotient.
  • The constant $K^2$ cannot be lowered to $(1-\varepsilon)K^2$, so the quadratic loss is a genuine feature of quotienting, not an artifact of the proof.
  • Without symmetry, the general bound is $K^3$ (or $K_1K_2$ for the two-sided doubling constants), improving the previously known $32K^6$ estimate and extending it from connected noncompact Lie groups to all unimodular locally compact groups.
  • Some subset $B$ carrying more than half the measure of $A$ has projection doubling below $2K$ (indeed $\alpha K$ for any $\alpha>1$), so almost all of $A$ behaves like a set with near-linear quotient doubling.
  • The mixed $K_1K_2$ bound gives a Ruzsa-distance formulation for nonsymmetric quotient doubling that may be a useful invariant in further work.

Reading between the lines

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

  • The sharpness example suggests a natural test: construct a nonsymmetric variant of the Cantor-set example; if its quotient doubling approaches $K^3$, the paper's general bound is sharp, whereas if it stays near $K^2$, the authors' conjecture that $K^2$ holds without symmetry gains support.
  • Because the proof uses only the metric properties of the Ruzsa distance and fiber slicing, the same arguments may transfer to other doubling notions, such as metric entropy or Banach density, where quotient versions are currently lacking.
  • The set-theoretic absoluteness remark implies the main theorem is a fully constructive statement about Borel sets, so effective or computable versions of the bound could be extracted by following the $\sigma$-compact approximation route.
  • The atom-dependence of the sharpness construction indicates the universal statement 'any compact group' in Proposition 2.1 should be read as 'any compact group with non-atomic Haar measure'; checking atomic cases would clarify the precise extent of sharpness.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. This paper studies how small-doubling sets behave under quotient maps in unimodular locally compact groups. Theorem 1.1 states that if A = A^{-1} is compact with μ_G(A^2) ≤ Kμ_G(A), H is a closed normal subgroup, and π: G → G/H is the quotient map, then μ_{G/H}((πA)^2) ≤ K^2 μ_{G/H}(πA); it further claims that the exponent 2 is optimal, in the sense that K^2 cannot be replaced by (1−ε)K^2 for any ε > 0. Proposition 2.1 supplies the sharpness example. Theorem 4.2 proves the general (non-symmetric) bound μ_{G/H}((πA)^2) ≤ K^3 μ_{G/H}(πA), improving on an earlier result of An, Jing, Zhang, and the third author, and a refined K_1 K_2 bound when both A^2 and A^{-1}A have at most K_1- and K_2-fold doubling; Theorem 1.2, generalized as Theorem 5.1, shows that for every α > 1 there is a compact B ⊆ A with μ_G(B) > (α−1)μ_G(A)/α and μ_{G/H}((πB)^2) < αK μ_{G/H}(πB). The proofs combine fiber-length functions, the layer-cake formula, a spillover inequality from [JT20], σ-compact modifications of superlevel sets, and the Ruzsa triangle inequality.

Significance. The main upper-bound arguments appear sound: I checked the layer-cake identities, the Hölder step in (4.2.3), the Ruzsa triangle passages, and the contradiction arguments in Theorem 4.2, and they are internally consistent and non-circular, with [JT20] and [Tao08] used as independent external facts. If the two load-bearing points flagged below are repaired, this is a strong contribution: the K^2 bound is sharp (the example attains quotient doubling K^2 − 2K + 2), it substantially improves the previous qualitative 32K^6 bound of [AJTZ21], and the K_1K_2 refinement and the large-subset theorem are natural and useful. The σ-compact modification machinery of Lemma 3.2 is a clean way to handle the measurability obstructions, and the paper is honest about where details are skipped. The two issues — the overstated claim in Proposition 2.1 that H may be any compact group, and the unproved compact-approximation step in the proof of Theorem 5.1 — are local and repairable, but each currently sits at a load-bearing position for a stated part of the main theorems.

major comments (2)
  1. [Section 2, Proposition 2.1] The stated claim that H may be an arbitrary compact group is false, and the measure computation omits the product term A_2^2. Since C + C = T, one has A_2^2 = {1_H} × M^2 × T, with μ_G(A_2^2) = μ_H({1_H})·|M^2|; moreover M^2 ⊄ I ∪ M (e.g., M_i M_j has determinant 1 for i ≠ j, while every element of I ∪ M has determinant ±1 with the non-identity elements having determinant −1). Therefore A_2^2 is not contained in the union A_1^2 ∪ A_1A_2 ∪ A_2A_1, and for finite H, for which the normalized Haar measure gives μ_H({1_H}) = 1/|H| > 0, the computation yields μ_G(A^2) = Kμ_G(A) + |M^2 ∖ (I ∪ M)|/|H| > Kμ_G(A), contradicting the proposition's conclusion. The calculation is valid precisely when H has no atoms, e.g., H = T. Since the sharpness assertion of Theorem 1.1 requires only one example per K, the conclusion survives by taking H = T, but the proposition must be restated and the vanishing of the omitted term justified.
  2. [Section 5, proof of Theorem 5.1] The final step of the proof, in both the case inf S = 0 and the case inf S > 0, asserts that one can approximate A_s from inside by a compact set B with μ_G(B) > (α−1)μ_G(A)/α and μ_{G/H}((πB)^2) < αK μ_{G/H}(πB). This passage is not justified as written: there is no reason that a compact approximation of A_s has πB close in measure to πA_s, because the fiber lengths of A_s over πA_s are only bounded below (by s) and can be unbounded above, so μ_G(B) close to μ_G(A_s) does not force μ_{G/H}(πB) close to μ_{G/H}(πA_s); consequently the strict inequality μ_{G/H}((πA_s)^2) < αK μ_{G/H}(πA_s) does not pass to arbitrary compact subsets. The step is repairable: choose a compact L ⊆ πA_s with μ_{G/H}(L) > μ_{G/H}((πA_s)^2)/(αK) and set B = π^{-1}(L) ∩ A, which is compact when A is compact; then πB = L, so μ_{G/H}((πB)^2) = μ_{G/H}(L^2) ≤ μ_{G/H}((πA_s)^2) < αK μ_{G/H}(L) = αK μ_{G/H}(πB), and L can be taken close enough to πA_s so that μ_G(B) > (α−1)μ_G(A)/α; for σ-compact A, an additional inner-regularity limit along increasing compact subsets works because μ_{G/H}(πK_n) increases to μ_{G/H}(πB). As written, however, the proof is incomplete at a point that supports Theorem 1.2.
minor comments (6)
  1. [Section 4, proof of Theorem 4.2(iii)] The displayed chain 'log(K1K2) ≥ log(μ(πAπA_t)/μ(πA_t)) + log(μ(πA_t^{-1}πA)/μ(πA_t)) = d(πA, πA_t^{-1}) + d(πA_t^{-1}, πA^{-1}) ≥ d(πA, πA^{-1})' contains a false equality: the middle sum exceeds the sum of the two Ruzsa distances by log(μ(πA)/μ(πA_t)) ≥ 0. Replacing '=' by '≥' gives the valid chain log(K1K2) ≥ [sum] ≥ d(πA, πA_t^{-1}) + d(πA_t^{-1}, πA^{-1}) ≥ d(πA, πA^{-1}), so the intended conclusion μ((πA)^2) ≤ K1K2 μ(πA) follows; this is a typographical slip rather than a substantive error.
  2. [Section 2, Proposition 2.1] The set S = {2^k : k ∈ [N]} and the observation |S − S| = N(N−1)+1 are never used in the proof; the identity |(I ∪ M)^2| = 4N^2 + 1 is asserted without derivation. Either supply the short verification or delete the unused set S.
  3. [Remark 4.3] The claim that Shoenfield absoluteness implies Theorem 4.2 without the Axiom of Choice is only sketched: one would need to spell out the reduction to Lie groups via Gleason–Yamabe and to verify the complexity assertion for a statement quantifying over locally compact groups and Haar measures. Since the main proof does not use this remark, it could be shortened or moved to a footnote.
  4. [References] The bibliography needs cleanup: [JT20] is cited as 2020 in the text but carries arXiv number 2303.15628; [BG08b] and [BG08c] are exact duplicates; [Hru20] and [Hru22] appear to be the same paper; and [HRF0] has an unusual citation format that should be resolved.
  5. [Section 4, proof of Theorem 4.2(i)] The equality Kμ(A) = ∫ Kμ(πA_t)dt = ∫ Kμ(~πA_t)dt is stated with only a reference to Lemma 3.2; it holds because for every t > 0, the difference πA_t minus ~πA_t is contained in the difference πA_t minus ~πA_{t/2}, which is null by Lemma 3.2(iv), so μ(πA_t) = μ(~πA_t) for all t > 0. A sentence of justification would help the reader.
  6. [Section 2, first paragraph] There is a typo in 'the chosen Haar measure is chosen is the product measure'; in addition, the abstract could remind the reader that πA^2 abbreviates (πA)^2, as defined in Section 1.2.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: Theorem 4.2 is proved from the spillover inequality (re-proved here) and standard Ruzsa distance facts; the sharpness example has a non-circular measure-theoretic gap for atomic H.

full rationale

The paper's central upper bound (Theorem 4.2/Theorem 1.1) does not assume the target inequality. The proof combines Lemma 3.2 (a σ-compact modification), Lemma 3.3 (spillover inequality), and the Ruzsa distance triangle inequality from [Tao08]. Lemma 3.3 is proved in the paper from the definition of fiber length, not imported as an unexamined black box; the citation to [JT20] is only for the name of the technique. The self-citation [AJTZ21] is used to describe the previously known K^6 bound being improved, and the proof does not rely on that bound. No fitted parameter is renamed as a prediction: the sharpness example in Proposition 2.1 is a construction. However, that construction has a correctness gap: when A=A1⊔A2, the computation μ_G(A^2)=μ_G(A1^2∪A1A2∪A2A1) omits A2^2={1_H}×M^2×C^2, whose measure is μ_H({1_H})·|M^2|·μ_T(C^2). For finite or atomic compact H, μ_H({1_H})>0 and the equality μ_G(A^2)=K fails, so Proposition 2.1 as stated for 'any compact group' is false; choosing H infinite, for example H=T, makes μ_H({1_H})=0 and repairs the sharpness example. This is a correctness risk, not circularity, because the sharpness claim is independent of the target upper-bound derivation and the K^2 lower bound can be exhibited with a valid H. Overall, the derivation chain is self-contained and not circular; score 1 reflects only minor non-load-bearing self-citations.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim adds no free parameters and no postulated entities. It inherits standard Haar measure theory, Ruzsa distance facts, and the spillover technique from prior literature.

assumptions (4)
  • domain assumption G, H, and G/H are unimodular and Haar measures are normalized so that Fubini's theorem holds in the fiber integral form.
    Section 1.2 convention; this makes μ_H bi-invariant and lets fiber lengths behave as needed throughout Theorems 4.2 and 5.1.
  • standard math All considered sets are compact or σ-compact so that Haar measurability and inner regularity apply, and product sets of measurable sets may be modified via σ-compact subsets.
    Used in Lemmas 3.1 through 3.3 and throughout Section 4; this follows from standard Haar measure theory.
  • standard math The Ruzsa distance on σ-compact sets satisfies nonnegativity, symmetry, the triangle inequality, and translation invariance.
    Fact 4.1 is quoted from Tao [Tao08] and is used at the end of the proof of Theorem 4.2(i) and (iii).
  • standard math The layer-cake identity ∫ μ_{G/H}(πA_t)dt = μ_G(A) holds for the fiber superlevel sets of a measurable set A.
    Lemma 3.1; this is the standard layer-cake representation for the measurable fiber length function.

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Pith. "Pith review of Measure doubling in unimodular locally compact groups and quotients." pith.science (2026). https://pith.science/paper/43AV5TVC

@misc{pith2026241117246,
  author       = {Pith},
  title        = {Pith review of: Measure doubling in unimodular locally compact groups and quotients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/43AV5TVC}},
  note         = {Machine review of arXiv:2411.17246}
}
abstract

We consider a (possibly discrete) unimodular locally compact group $G$ with Haar measure $\mu_G$, and a compact $A\subseteq G$ of positive measure with $\mu_G(A^2)\leq K\mu_G(A)$. Let $H$ be a closed normal subgroup of G and $\pi: G \rightarrow G/H$ be the quotient map. With the further assumption that $A= A^{-1}$, we show $$\mu_{G/H}(\pi A ^2) \leq K^2 \mu_{G/H}(\pi A).$$ We also demonstrate that $K^2$ cannot be replaced by $(1-\epsilon)K^2$ for any $\epsilon>0$. In the general case (without $A=A^{-1}$), we show $\mu_{G/H}(\pi A ^2) \leq K^3 \mu_{G/H}(\pi A)$, improving an earlier result by An, Jing, Zhang, and the third author. Moreover, we are able to extract a compact set $B\subseteq A$ with $\mu_G(B)> \mu_G(A)/2$ such that $ \mu_{G/H}(\pi B^2) < 2K \mu_{G/H}(\pi B)$.

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