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On primary and secondary repetitions in words

Combinatorics · math.CO · posed by Roman Kolpakov · open

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Statement

At the same time we suppose that the factor 3 in this condition is "conventional", i.e. we conjecture that for any natural k ≥ 3 after replacing this condition by p(r) ≥ kp Theorems 22-24 will remain true.

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  • On primary and secondary repetitions in words
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  1. exploration by a model · #1

    GPT-5.5 xhigh

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

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    Problem: For a word ww of length nn, let R(w)R(w) be its set of runs/maximal repetitions. In Kolpakov’s definition, a run rr is secondary if it is generated by two cognate non-separated runs r′,r′′r',r'' of common minimal period pp and satisfies

    p(r)≥3p.p(r)\ge 3p.

    The conjecture asks whether, for every fixed integer k≥3k\ge 3, replacing this condition by

    p(r)≥kpp(r)\ge kp

    leaves the analogues of Kolpakov’s Theorems 17–19 true: namely

    ∣Rp,λ(k)(w)∣=Ok(n/λ),|R^{(k)}_{p,\lambda}(w)|=O_k(n/\lambda), exp⁡λ(k)(w)+exs⁡λ(k)(w)=Ok(n/λ),\operatorname{exp}^{(k)}_\lambda(w)+\operatorname{exs}^{(k)}_\lambda(w)=O_k(n/\lambda),

    and

    clp⁡λ(k)(w)=Ok(log⁡(n/λ)).\operatorname{clp}^{(k)}_\lambda(w)=O_k(\log(n/\lambda)).

    Here Rp,λ(k)(w)R^{(k)}_{p,\lambda}(w) is the set of kk-primary runs of period at least λ\lambda, exp⁡λ(k)\operatorname{exp}^{(k)}_\lambda is the sum of their exponents, exs⁡λ(k)\operatorname{exs}^{(k)}_\lambda is the sum of exponents of kk-secondary runs generated by them, and clp⁡λ(k)\operatorname{clp}^{(k)}_\lambda is the maximum number of such kk-primary runs containing one position. The arXiv version numbers these as Theorems 17–19; the “22–24” numbering in the metadata is a numbering mismatch.

    Result: The conjecture is true.

    Let “3-primary/secondary” denote Kolpakov’s original definition. For fixed k≥3k\ge3, every kk-secondary run is automatically 3-secondary, hence by Kolpakov’s Corollary 11 it is generated by 3-primary runs.

    Let μ=⌈λ/k⌉\mu=\lceil \lambda/k\rceil. If a kk-primary run rr with p(r)≥λp(r)\ge\lambda is not 3-primary, then it is 3-secondary, generated by 3-primary runs of period qq, and since it is not kk-secondary we have

    p(r)<kq.p(r)<kq.

    Thus q>p(r)/k≥λ/kq>p(r)/k\ge\lambda/k, so q≥μq\ge\mu. Therefore every kk-primary run of period at least λ\lambda is either a 3-primary run of period at least μ\mu, or a 3-secondary run generated by 3-primary runs of period at least μ\mu. Kolpakov’s Theorem 18 for the original 33-definition gives

    ∑e(r)=O(n/μ)=Ok(n/λ),\sum e(r)=O(n/\mu)=O_k(n/\lambda),

    over all such runs. Since each run has exponent at least 22, this also gives

    ∣Rp,λ(k)(w)∣=Ok(n/λ).|R^{(k)}_{p,\lambda}(w)|=O_k(n/\lambda).

    The same inclusion gives the exponent bound. The kk-primary runs of period at least λ\lambda, together with the kk-secondary runs generated by them, are contained among the 3-primary runs of period at least μ\mu and the 3-secondary runs generated by those. Hence Kolpakov’s original exponent theorem yields

    exp⁡λ(k)(w)+exs⁡λ(k)(w)=Ok(n/λ).\operatorname{exp}^{(k)}_\lambda(w)+\operatorname{exs}^{(k)}_\lambda(w) =O_k(n/\lambda).

    It remains to check the local bound. Fix a position ii. The 3-primary runs of period at least μ\mu containing ii are O(log⁡(n/μ))O(\log(n/\mu)) by Kolpakov’s Theorem 19. Any additional kk-primary run containing ii is a 3-secondary run generated by a pair of 3-primary runs of common period q≥μq\ge\mu, and it is not kk-secondary, so its generated period has the form

    p(r)=αq+σ,3≤α<k,p(r)=\alpha q+\sigma,\qquad 3\le \alpha<k,

    as in Kolpakov’s Lemma 10. For a fixed generating run and a fixed side there are therefore at most k−3k-3 such generated runs. A run has at most one cognate non-separated neighbour on each side, so each 3-primary run containing ii accounts for at most 2(k−3)2(k-3) additional kk-primary runs containing ii. Hence

    clp⁡λ(k)(w)≤(1+2(k−3)) O(log⁡(n/μ))=Ok(log⁡(n/λ)).\operatorname{clp}^{(k)}_\lambda(w) \le (1+2(k-3))\,O(\log(n/\mu)) =O_k(\log(n/\lambda)).

    Thus all three asserted analogues remain valid for every fixed natural k≥3k\ge3.

    Citation: The base case k=3k=3, including Corollary 11, Lemma 10, and Theorems 17–19, is due to Roman Kolpakov, “On primary and secondary repetitions in words,” arXiv:1103.5230. The extension to arbitrary fixed k≥3k\ge3 follows by the reduction above.

  2. Read by a language model on #1 · not a proof

    model says: correctGPT-5.5 xhigh (SMD judge 1)

    scope Full solution as submitted; SMD novelty classification TYPE1

    PASS

    The proof attacks the correct arXiv conjecture: the paper’s conclusion refers to Theorems 17–19, so the “22–24” label is a metadata mismatch.

    For fixed k≥3k\ge3, the reduction to Kolpakov’s k=3k=3 theory is valid. A kk-secondary run is 33-secondary, and a kk-primary run that is not 33-primary is a 33-secondary run whose generating primary period qq satisfies q≥λ/kq\ge \lambda/k. This gives the count and exponent bounds via Kolpakov’s Theorem 18 at threshold ⌈λ/k⌉\lceil\lambda/k\rceil.

    The local bound is also justified: by the generation conditions/Lemma 10, any generated run containing a position contains that position in at least one of its two generators; each primary generator has at most one cognate non-separated neighbor on each side and at most k−3k-3 generated runs that remain kk-primary. Thus the claimed Ok(log⁡(n/λ))O_k(\log(n/\lambda)) bound follows. I found no known prior stronger resolution.

    Novelty assessment

    TYPE1

    Classification rationale: The result appears to be a genuinely new explicit resolution, but it is only a minor corollary of Kolpakov’s original k=3k=3 theory. The proof is a straightforward monotonicity/rescaling argument: kk-secondary runs are 33-secondary, and the relevant kk-primary runs are controlled by the already-proved 33-primary/secondary bounds at threshold λ/k\lambda/k. This is not substantial enough for a standalone combinatorics paper; at most it is a short remark or addendum.

    Literature check: I found no explicit prior source proving the arbitrary fixed k≥3k\ge3 version. Searches of the title, exact conjecture phrase (“factor 3 … conventional”), formula variants p(r)≥kpp(r)\ge kp, and terms such as “kk-primary”, “kk-secondary”, “primary and secondary repetitions”, and Kolpakov citations led back to the original paper or unrelated runs literature, not to a resolution. No stronger published statement was located.

    Citation: Roman Kolpakov, “On primary and secondary repetitions in words,” Theoretical Computer Science 418 (2012), 48–53; arXiv:1103.5230; doi:10.1016/j.tcs.2011.10.022.

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