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DISCRETE EQUIDECOMPOSABILITY AND EHRHART THEORY OF POLYGONS

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discrete-equidecomposability-and-ehrhart-theory-of-polygonsCombinatoricsmath.COposed by Paxton Turner, Yuhuai (Tony) Wurecorded: open · 1 machine check, unexamined

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Statement

Ehrhart equivalence is a necessary and sufficient condition for (not necessarily finite or rational) discrete equidecomposability.

Context

Candidate 1 of the open problems stated in "DISCRETE EQUIDECOMPOSABILITY AND EHRHART THEORY OF POLYGONS", 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: Reconstructed statement (Turner--Wu Conj. 1.6, made precise as their later Conj. 4.6): for bounded subsets S,S′⊂R2S,S'\subset \mathbb R^2, allowing arbitrary not-necessarily-finite and not-necessarily-rational Δ\Delta-complex decompositions, SS and S′S' are discretely equidecomposable by piecewise maps in

    G=GL2(Z)⋉Z2G=GL_2(\mathbb Z)\ltimes \mathbb Z^2

    if and only if

    ehr⁡S(t)=∣tS∩Z2∣=∣tS′∩Z2∣=ehr⁡S′(t)∀t∈Z>0.\operatorname{ehr}_S(t)=|tS\cap \mathbb Z^2|=|tS'\cap \mathbb Z^2|=\operatorname{ehr}_{S'}(t) \quad\forall t\in\mathbb Z_{>0}.

    Result: The statement is false, even for rational line segments.

    Let

    E1=conv⁡{(1/5,0),(0,1/5)},E3=conv⁡{(2/5,0),(1/5,1/5)}.E_1=\operatorname{conv}\{(1/5,0),(0,1/5)\},\qquad E_3=\operatorname{conv}\{(2/5,0),(1/5,1/5)\}.

    Then E1E_1 lies on x+y=1/5x+y=1/5, while E3E_3 lies on x+y=2/5x+y=2/5.

    For t≥1t\ge1, tE1tE_1 has lattice points iff 5∣t5\mid t. If t=5kt=5k, then the lattice points satisfy X+Y=kX+Y=k, giving k+1k+1 points. Similarly tE3tE_3 has lattice points iff 5∣t5\mid t, and for t=5kt=5k the points are (2k−j,j)(2k-j,j), 0≤j≤k0\le j\le k, again k+1k+1 points. Hence

    ehr⁡E1(t)=ehr⁡E3(t)∀t≥1.\operatorname{ehr}_{E_1}(t)=\operatorname{ehr}_{E_3}(t) \quad\forall t\ge1.

    Now suppose an arbitrary discrete equidecomposition existed. Every point must be sent by some GG-map to a point in its own GG-orbit. We show that uncountably many points of E1E_1 have no GG-translate in E3E_3.

    Parametrize

    ps=(s,1/5−s),0≤s≤1/5.p_s=(s,1/5-s),\qquad 0\le s\le 1/5.

    Let g(x)=Ax+v∈Gg(x)=Ax+v\in G, with

    A=(abcd)∈GL2(Z),v=(m,n)∈Z2.A=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in GL_2(\mathbb Z),\qquad v=(m,n)\in\mathbb Z^2.

    If g(ps)∈E3g(p_s)\in E_3, then the coordinate sum of g(ps)g(p_s) equals 2/52/5. This gives

    ((a+c)−(b+d))s+b+d5+m+n=25.\bigl((a+c)-(b+d)\bigr)s+\frac{b+d}{5}+m+n=\frac25.

    If (a+c)−(b+d)≠0(a+c)-(b+d)\ne0, this equation has at most one solution ss. If instead a+c=b+d=ra+c=b+d=r, then the equation requires

    r+5(m+n)=2,r+5(m+n)=2,

    so r≡2(mod5)r\equiv2\pmod5. But since the two column sums of AA are both rr,

    det⁡A=r(a−b),\det A=r(a-b),

    and det⁡A=±1\det A=\pm1, forcing r=±1r=\pm1, impossible modulo 55. Thus for each g∈Gg\in G, at most one ss satisfies g(ps)∈E3g(p_s)\in E_3.

    Since GG is countable, only countably many s∈[0,1/5]s\in[0,1/5] have Gps∩E3≠∅G p_s\cap E_3\ne\varnothing. Choose s0s_0 outside this countable set. Then ps0∈E1p_{s_0}\in E_1, but its GG-orbit does not meet E3E_3. No piecewise-GG bijection E1→E3E_1\to E_3 can therefore exist.

    Thus Ehrhart equivalence is not sufficient for arbitrary discrete equidecomposability.

    Citation: The conjecture and the two motivating segments appear in Turner--Wu, Discrete Equidecomposability and Ehrhart Theory of Polygons, arXiv:1412.0196. The counterexample above is self-contained.

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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 is mathematically sound for the supplied broad formulation allowing arbitrary bounded subsets/decompositions. The two rational segments have identical Ehrhart counts for every positive dilation. The orbit obstruction is rigorous: for each affine unimodular g∈GL2(Z)⋉Z2g\in GL_2(\mathbb Z)\ltimes\mathbb Z^2, at most one point of E1E_1 can map into E3E_3; since GG is countable, only countably many points of E1E_1 can have any GG-translate in E3E_3. Thus no piecewise-GG bijection/equidecomposition can exist. This disproves sufficiency of Ehrhart equivalence in the stated arbitrary setting.

      Novelty assessment

      TYPE1

      Classification rationale: The counterexample is mathematically neat but very small: it exploits the elementary fact that any piecewise GL2(Z)⋉Z2GL_2(\mathbb Z)\ltimes\mathbb Z^2 map preserves pointwise GG-orbits, while Ehrhart counts see only lattice points in dilates. It falsifies the broad “arbitrary/not necessarily rational or finite” formulation, but only via one-dimensional segments and does not substantially address the intended polygonal/equidecomposition program. This is best viewed as a short observation or caveat, not a standalone publishable combinatorics paper.

      Literature check: I found no prior source explicitly giving this orbit-obstruction counterexample or a stronger disproof of the broad arbitrary-equidecomposability version. Searches of arXiv for “discrete equidecomposability” found only Turner–Wu’s original paper and their companion paper on finite rational equidecomposability. The companion paper gives conditions for rational finite equidecomposability, not this arbitrary/infinite line-segment counterexample. Searches for the exact terminology, the conjecture wording, the segment coordinates, “Ehrhart equivalence” with “discrete equidecomposability,” GitHub repositories/issues/discussions, and related open web sources did not reveal a known resolution. The closest prior material is Turner–Wu’s own use of these denominator-5 edge examples/weight obstructions, but not the uncountable-orbit argument ruling out arbitrary piecewise-GG equidecomposition of the two segments.

      Citation: Paxton Turner and Yuhuai Wu, “Discrete Equidecomposability and Ehrhart Theory of Polygons,” arXiv:1412.0196. Related: Turner and Wu, “Conditions for Discrete Equidecomposability of Polygons,” arXiv:1412.0191.

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