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Canonical versions of van der Waerden and Szemerédi type recurrence (conjectures and implications)

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canonical-versions-of-van-der-waerden-and-szemeredi-type-recurrence-2Number Theorymath.MGmath.NTposed by Unknownrecorded: open · 1 machine check, unexamined

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Statement

Conjecture 1 holds if X is assumed to be a compact metric space.

Context

Candidate 2 of the open problems stated in "Canonical versions of van der Waerden and Szemerédi type recurrence (conjectures and implications)", extracted for the Scalable Mathematical Discovery run.

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  • #1

    Attempt 1

    ai explorationGPT-5.5 xhigh ·
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    ai discovered
    — the result was found by a model.
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    NEW

    Problem: Reconstructed Conjecture 2: with N={1,2,… }\mathbb N=\{1,2,\dots\}, for every compact metric space XX, continuous map T:X→XT:X\to X, open set U⊆XU\subseteq X, and ℓ∈N\ell\in\mathbb N, there exists n≥1n\ge1 such that either

    U∩T−nU∩⋯∩T−ℓnU≠∅,U\cap T^{-n}U\cap\cdots\cap T^{-\ell n}U\neq\varnothing,

    or

    T−inU∩T−jnU=∅(0≤i<j≤ℓ).T^{-in}U\cap T^{-jn}U=\varnothing\qquad(0\le i<j\le \ell).

    This is exactly Conjecture 1 with the additional hypothesis “XX compact metric.”

    Result: The conjecture is false already for ℓ=2\ell=2.

    Let

    A={∑k=0rεk3k:εk∈{0,1}}⊆N0.A=\Bigl\{\sum_{k=0}^r \varepsilon_k3^k:\varepsilon_k\in\{0,1\}\Bigr\}\subseteq\mathbb N_0 .

    Then AA has no nontrivial 3-term arithmetic progression: if a,c,b∈Aa,c,b\in A and a+c=2ba+c=2b, comparing ternary digits modulo 33 recursively forces all digits of a,b,ca,b,c equal.

    Also A−A=ZA-A=\mathbb Z, because every integer has a finite balanced ternary expansion with digits in {−1,0,1}\{-1,0,1\}.

    Let x∈{0,1}Zx\in\{0,1\}^{\mathbb Z} be the indicator of AA, let X={σmx:m∈Z}‾X=\overline{\{\sigma^m x:m\in\mathbb Z\}}, where σ\sigma is the shift (σy)k=yk+1(\sigma y)_k=y_{k+1}. Then XX is compact metric and σ:X→X\sigma:X\to X is continuous. Let

    U={y∈X:y0=1}.U=\{y\in X:y_0=1\}.

    For every n≥1n\ge1, since n∈A−An\in A-A, choose a,b∈Aa,b\in A with a−b=na-b=n. Then σbx∈U∩σ−nU\sigma^b x\in U\cap \sigma^{-n}U, so the pairwise-disjoint alternative fails.

    On the other hand, if

    U∩σ−nU∩σ−2nU≠∅,U\cap\sigma^{-n}U\cap\sigma^{-2n}U\neq\varnothing,

    then some orbit point σmx\sigma^m x has 1’s in coordinates 0,n,2n0,n,2n, so m,m+n,m+2n∈Am,m+n,m+2n\in A, a nontrivial 3-term arithmetic progression, impossible. Thus the recurrence alternative also fails.

    Therefore no nn satisfies the proposed dichotomy.

    Citation: No external resolution is used; the counterexample is elementary.

    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 proposed counterexample is mathematically valid for the stated reconstructed dichotomy. The set AA of integers with ternary digits only 0,10,1 is 3-term-AP-free and satisfies A−A=ZA-A=\mathbb Z. The associated two-sided subshift is compact metric, the cylinder U={y:y0=1}U=\{y:y_0=1\} is open, and for every n≥1n\ge1 the pairwise-disjoint alternative fails because n∈A−An\in A-A. The recurrence alternative would force a length-3 arithmetic progression in AA; in the orbit-closure setting this follows since the relevant cylinder is open, so the finite pattern must occur in some shift of xx. Thus no nn satisfies the dichotomy, disproving the conjecture for ℓ=2\ell=2.

      Novelty assessment

      TYPE1

      Classification rationale: The counterexample is valid but mathematically routine: it combines the standard Stanley/Salem–Spencer sequence of integers with ternary digits only 0,10,1, balanced ternary giving A−A=ZA-A=\mathbb Z, and the standard subshift/cylinder encoding. It is a useful correction to the conjecture, but too elementary and folklore-adjacent to support a standalone journal paper.

      Literature check: I found no source explicitly stating that Farhangi’s compact-metric canonical recurrence conjecture is false. Searches for the exact title/phrases and for variants involving “canonical Szemerédi,” “canonical van der Waerden recurrence,” “3-AP-free difference basis,” “Stanley sequence difference set,” and “balanced ternary” led only to standard sources for the ingredients. Those ingredients essentially imply the counterexample immediately.

      Citation: No direct prior citation for the exact topological counterexample found. Standard ingredients: OEIS A005836; Janusz Dybizbański, “Sequences containing no 3-term arithmetic progressions,” Electron. J. Combin. 19(2) (2012), P15; R. Salem and D. C. Spencer, “On sets of integers which contain no three terms in arithmetical progression,” PNAS 28 (1942), 561–563.

      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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