{"id":"de994f35-34c4-48ae-be47-f305d8f4c7af","arxiv_id":"2411.17246","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Symmetric compact sets with K-doubling have quotient doubling at most K^2; without symmetry, K^3 suffices.","lead":"A mathematics paper proves new bounds on how much a set can grow when you square it after passing to a quotient group. The symmetric case gives a sharp quadratic bound, and the nonsymmetric case gives a cubic bound.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2.1's sharpness example omits A2^2, so it is valid only for non-atomic compact H; the stated 'any compact group' is false, though choosing H=T repairs it.","rationale":"The central upper-bound theorems are supported by a coherent spillover/Ruzsa argument; I found no substantive error there. The serious blemish is in the sharpness example, where the product A2^2 is silently dropped; this is a genuine mathematical gap in the proof as written and matches the reader's weakest_assumption exactly. It is localized and repairable: the theorem's 'moreover' requires only existence, and H=T or any infinite compact group works. Consequently the verdict should remain CONDITIONAL, pending the repair. I also reviewed the sketch in Theorem 5.1 and the algebra in Theorem 4.2(iii); both are fixable and do not alter the assessment.","tokens_in":11377,"tokens_out":29762,"duration_ms":238277,"concrete_test":"Run the construction with N=1 (K=3), H=Z/2Z with normalized counting measure, and M={[[0,1],[1,2]],[[−2,1],[1,0]]}. The omitted A2^2={1}×M^2×T has |M^2|=3 and |M^2∩(I∪M)|=1 (only I), so μ_G(A^2)=K+μ_H({1})(|M^2|−1)=3+1/2·2=4>Kμ_G(A), contradicting Proposition 2.1 as stated. Repeat with H=T: μ_H({1})=0, so μ_G(A^2)=3 and μ_Q(πA^2)=5=K^2−2K+2, confirming sharpness after repair.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 2.1 defines A=A1⊔A2 with A1=H×I×T and A2={1_H}×M×C, then computes μ_G(A^2) as the measure of A1^2∪A1A2∪A2A1. The omitted term A2^2={1_H}×M^2×T has measure μ_H({1_H})·|M^2|·μ_T(T). For finite compact H, μ_H({1_H})>0, and since M^2 is not contained in I∪M, A2^2 contributes positive measure beyond the union; hence μ_G(A^2)>Kμ_G(A), contrary to the proposition. Thus the proof of the sharpness assertion in Theorem 1.1 is incomplete as written, and the universal claim 'H any compact group' is false. The sharpness conclusion itself survives, because it only requires one example: taking H=T or any infinite compact group makes μ_H({1})=0, so the omitted term vanishes and the computation μ_G(A^2)=K, μ_Q(πA^2)=K^2−2K+2 goes through. The upper-bound arguments in Section 4 appear mathematically sound; the only other blemish is a typographical equality in the proof of Theorem 4.2(iii) that should be an inequality, but the intended Ruzsa conclusion still follows.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":11646,"tokens_out":46678,"duration_ms":341135,"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":[{"comment":"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.","section":"Section 2, Proposition 2.1"},{"comment":"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.","section":"Section 5, proof of Theorem 5.1"}],"minor_comments":[{"comment":"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.","section":"Section 4, proof of Theorem 4.2(iii)"},{"comment":"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.","section":"Section 2, Proposition 2.1"},{"comment":"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.","section":"Remark 4.3"},{"comment":"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.","section":"References"},{"comment":"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.","section":"Section 4, proof of Theorem 4.2(i)"},{"comment":"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.","section":"Section 2, first paragraph"}],"recommendation":"major_revision","confidential_remarks":"Both load-bearing issues identified in the report are local and repairable, and I do not doubt that the main theorems are correct after the repairs: the sharpness construction works verbatim with H = T, and the final step of Theorem 5.1 can be fixed by the compact-L construction described in the major comment. The upper-bound half of the paper (Theorem 4.2) is, in my reading, correct and clean. The paper fits the journal's scope well; its selling points are the sharp K^2 symmetric quotient bound, the sharpness example, and the large-subset corollary. I would suggest the editor treat the revision as straightforward but insist that Proposition 2.1's hypothesis be corrected and the compact-approximation argument in Theorem 5.1 be written out. Housekeeping: duplicate references and the sketched absoluteness remark."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here are my honest thoughts on arXiv:2411.17246. The main theorems are genuinely new and the paper is worth engaging: it sharpens the quotient doubling bound from 32K^6 in [AJTZ21] to K^2 for symmetric sets and K^3 in general, and extends the result from connected Lie groups to all unimodular locally compact groups. There's also a nice large-subset theorem with linear doubling. The upper-bound proof is sound; the spillover technique is prior work, but the way it's applied here—via sigma-compact approximations and Ruzsa triangle—is clean and correct. I checked the Ruzsa steps and the contradiction argument in Theorem 4.2; they hold.\n\nThe soft spot is in the sharpness example, Proposition 2.1. The measure computation silently drops the A2^2 term. As written, mu_G(A^2) is computed as the measure of A1^2 union A1A2 union A2A1, but A2^2 = {1_H} x M^2 x T is not included. For finite or any atomic compact H, mu_H({1_H})>0, so that term has positive measure and the equality mu_G(A^2)=K is false. So the claim that this works for 'any compact group H' is overbroad. The fix is simple: require H to be non-atomic, e.g., H=T. Since you only need one example to prove sharpness, the theorem survives. This is an oversight in an example, not a flaw in the main argument. There is also a typo in the proof of Theorem 4.2(iii) where an equality should be an inequality, but the intended Ruzsa conclusion still follows.\n\nNet: the upper-bound results appear correct, the sharpness example is repairable, and the paper is a solid contribution to the small-doubling program. I'd send it to a serious referee. I would also bring it to the reading group.","headline":"Sharpness construction has a fixable gap, but the main quotient doubling bounds are new, correct, and worth refereeing.","tokens_in":12201,"tokens_out":2657,"would_cite":true,"duration_ms":22542,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22D05","51F99","22E30","03C99","11B30"],"pacs":[],"model":"deepseek-v4-flash","headline":"A compact symmetric set with doubling $K$ projects to quotient groups with doubling at most $K^2$, and this constant is optimal.","keywords":["measure doubling","locally compact groups","Haar measure","quotient groups","Ruzsa distance","spillover technique","small doubling","unimodular groups"],"falsifier":"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.","tokens_in":11176,"feed_emoji":"📐","tokens_out":15703,"duration_ms":118156,"temperature":0.7,"pith_summary":"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.","feed_headline":"Symmetric set doubling survives quotients with only K^2 loss","feed_subtitle":"Improves the known K^6-type bound to K^2 and proves no smaller constant works.","key_machinery":"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.","core_discovery":"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 $K^2$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"It provides the earlier $32K^6$ quotient bound for connected noncompact Lie groups that Theorem 1.1 improves, and it frames the small-measure-expansion problem.","marker":"[AJTZ21]"},{"why":"It introduces the spillover technique (Lemma 9.4) that Lemma 3.3 adapts to connect fiber superlevel sets with the measure of $AB$.","marker":"[JT20]"},{"why":"It provides the Ruzsa distance facts and triangle inequality used in the proof of Theorem 4.2.","marker":"[Tao08]"},{"why":"It documents the folklore obstruction to quotient doubling and motivates replacing doubling by entropic doubling, which this paper's sharp symmetric bound partially addresses.","marker":"[GGMT24]"}],"fun_headline_variants":["Symmetric doubling survives quotients: sharp K^2 loss","Quotient doubling sharp: K^2 for symmetric sets","Sharp quotient bound: doubling K becomes K^2","Optimal quotient doubling: K^2 from symmetric K-doubling"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Symmetric doubling survives quotients: sharp K^2 loss","Quotient doubling sharp: K^2 for symmetric sets","Sharp quotient bound: doubling K becomes K^2","Optimal quotient doubling: K^2 from symmetric K-doubling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000536,"raw_usage":{"total_tokens":2625,"prompt_tokens":1047,"completion_tokens":1578,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":663,"completion_tokens_details":{"reasoning_tokens":1507}},"tokens_in":663,"tokens_out":1578,"duration_ms":12606,"temperature":1.0,"reasoning_tokens":1507,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:22:08.185866+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}