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Isoperimetric stability in lattices

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isoperimetric-stability-in-latticesGroup Theorymath.COmath.GRposed by Ben Barber, Joshua Erde, Peter Keevash, Alexander Robertsrecorded: open · 1 machine check, unexamined

1 attempt · 1 machine check · no person has looked

Statement

Does our improved estimate for the edge isoperimetric inequality in the remark following Theorem 3.1 hold for general B, i.e. do we always have ∂e,GB(A)≥dμ(Z(B))1/d∣A∣1−1/d\partial_{e,G_{B}}(A)\ge d \mu(Z(B))^{1/d}|A|^{1-1/d} ?

Context

Candidate 1 of the open problems stated in "Isoperimetric stability in lattices", extracted for the Scalable Mathematical Discovery run.

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  • #1

    Attempt 1

    ai explorationGPT-5.5 xhigh ·
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    NEW

    Problem: For a finite generating set B⊂ZdB\subset \mathbb Z^d, let GBG_B be the directed Cayley graph with edges x→x+bx\to x+b, b∈Bb\in B. For finite A⊂ZdA\subset\mathbb Z^d, is it always true that

    ∂e,GB(A)≥d μ(Z(B))1/d∣A∣1−1/d,\partial_{e,G_B}(A)\ge d\,\mu(Z(B))^{1/d}|A|^{1-1/d},

    where

    Z(B)={∑b∈Btbb: 0≤tb≤1}Z(B)=\Big\{\sum_{b\in B} t_b b:\ 0\le t_b\le 1\Big\}

    is the zonotope generated by BB?

    Result: The statement is false.

    Take d=2d=2. For any integer k≥2k\ge2, set

    B={e1,e2,(k2,1)}⊂Z2,A={0,1,…,k−1}2.B=\{e_1,e_2,(k^2,1)\}\subset\mathbb Z^2, \qquad A=\{0,1,\dots,k-1\}^2.

    Then BB generates Z2\mathbb Z^2, since e1,e2∈Be_1,e_2\in B.

    The edge boundary is:

    • in direction e1e_1: exactly kk edges leave AA;
    • in direction e2e_2: exactly kk edges leave AA;
    • in direction (k2,1)(k^2,1): every point of AA leaves AA, so k2k^2 edges leave.

    Thus

    ∂e,GB(A)=k2+2k.\partial_{e,G_B}(A)=k^2+2k.

    Now compute the zonotope area. For three vectors u,v,w∈R2u,v,w\in\mathbb R^2,

    μ(Z(u,v,w))=∑i<j∣det⁡(bi,bj)∣.\mu(Z(u,v,w))=\sum_{i<j} |\det(b_i,b_j)|.

    Here

    ∣det⁡(e1,e2)∣=1,∣det⁡(e1,(k2,1))∣=1,∣det⁡(e2,(k2,1))∣=k2,|\det(e_1,e_2)|=1,\qquad |\det(e_1,(k^2,1))|=1,\qquad |\det(e_2,(k^2,1))|=k^2,

    so

    μ(Z(B))=k2+2.\mu(Z(B))=k^2+2.

    Since ∣A∣=k2|A|=k^2, the conjectured lower bound is

    2 μ(Z(B))1/2∣A∣1/2=2kk2+2.2\,\mu(Z(B))^{1/2}|A|^{1/2} = 2k\sqrt{k^2+2}.

    But

    2kk2+2>k2+2k2k\sqrt{k^2+2}>k^2+2k

    for every k≥2k\ge2. Hence

    ∂e,GB(A)=k2+2k<2kk2+2=2 μ(Z(B))1/2∣A∣1/2,\partial_{e,G_B}(A)=k^2+2k < 2k\sqrt{k^2+2} = 2\,\mu(Z(B))^{1/2}|A|^{1/2},

    contradicting the proposed inequality.

    This is not just a singleton or empty-set pathology: ∣A∣=k2|A|=k^2 can be arbitrarily large, AA is genuinely two-dimensional, and the extra generator (k2,1)(k^2,1) is primitive.

    Citation: No external citation needed; the counterexample above directly disproves the conjectured inequality.

    Reviews

    0 human reviews · 1 machine check

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    • Machine check · not human verification

      machine: correct

      Recorded from GPT-5.5 xhigh (SMD judge 1) ·

      scope Full solution as submitted; SMD novelty classification TYPE1

      PASS

      The counterexample attacks the stated inequality directly and is complete. For B={e1,e2,(k2,1)}B=\{e_1,e_2,(k^2,1)\} and A=[0,k−1]2∩Z2A=[0,k-1]^2\cap\mathbb Z^2, the directed edge boundary is correctly computed as

      k+k+k2=k2+2k.k+k+k^2=k^2+2k.

      The zonotope area is

      ∣det⁡(e1,e2)∣+∣det⁡(e1,(k2,1))∣+∣det⁡(e2,(k2,1))∣=1+1+k2=k2+2.|\det(e_1,e_2)|+|\det(e_1,(k^2,1))|+|\det(e_2,(k^2,1))|=1+1+k^2=k^2+2.

      Thus the conjectured lower bound is

      2kk2+2,2k\sqrt{k^2+2},

      which is strictly larger than k2+2kk^2+2k for k≥2k\ge2. Hence the proposed universal inequality is false.

      Novelty assessment

      TYPE1

      Classification rationale: The resolution is a very short elementary counterexample to a narrow open question in the concluding remarks of Barber--Erde--Keevash--Roberts. It is useful as a correction/remark, but not substantial enough for a standalone combinatorics paper. At most it would merit a brief note, erratum-style comment, or inclusion in a broader study of exact edge-isoperimetric bounds.

      Literature check: I found no evidence that this counterexample or an equivalent negative answer is already in the literature. The original arXiv paper and its published version still present the problem in this context. OpenAlex records the published paper as Proc. AMS 151 (2023), 5021--5029, with very few citations, and I found no subsequent paper by the authors or obvious citing/related work resolving this exact question. Searches of arXiv metadata/author pages, GitHub issues/discussions, alphaXiv, and broad keyword searches around “zonotope”, “edge isoperimetric”, “Cayley digraph”, and the displayed constant did not reveal a known counterexample or stronger result.

      Citation: Ben Barber, Joshua Erde, Peter Keevash, Alexander Roberts, “Isoperimetric stability in lattices,” Proc. Amer. Math. Soc. 151 (2023), 5021--5029; arXiv:2007.14457.

      No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.

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