| 研究課題/領域番号 |
19K11815
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| 研究機関 | 秋田大学 |
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研究代表者 |
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| 研究期間 (年度) |
2019-04-01 – 2022-03-31
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| キーワード | combinatorics on words / repetitions / squares / distinct squares / upper bounds |
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| 研究実績の概要 |
The main idea behind my approach to counting repetitions is to group them by their root and the partial order imposed on them by the prefix ordering. All repetitions whose roots share a common prefix are in one group. The aim is to show that for the shortest element in a group there are at least as many occurrences of its root as the cardinality of the group. This would imply the well-known distinct square conjecture of Fraenkel and Simpson.
Together with Robert Mercas, we proved our conjecture in the special case when the roots of repetitions considered form a chain (totally ordered set) with respect to the prefix ordering. The result does not directly generalize to partially ordered groups of repetition roots, but certain aspects of the proofs carry over to the more general case.
The manuscript summarizing the above results has been reviewed by colleagues and they found no mistakes in the proofs. We submitted it for a RIMS technical report and will submit it for peer reviewed publication in the near future.
I also managed to make the first steps toward unifying the treatment of upper bounds for distinct repetitions and maximal repetitions (runs): using the occurrence counting of the prefix-ordered roots, I showed that when the roots of runs form a single chain, the same upper bound applies as in the distinct square case.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
The project is progressing well. In the first year we managed to break through by proving the conjecture on the number of root occurrences in the special case when the roots form a linear order. This is the main achievement of the first year. We are confident that we can build on this foundation towards the proof of the upper bound given in the general conjecture.
We have also worked on lower bounds, i.e., constructing words which achieve high density of distinct repetitions, using the insight obtained by the main result. For roots forming a chain, we were able to show that any sequence of cluster size is attainable. This direction will also be further pursued towards lower bounds in the general case, as planned.
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| 今後の研究の推進方策 |
I plan to continue the work according mainly to the original plan, that is, focusing most of the effort on generalizing the bounds obtained in the first year. In order to do this, the first step will be to consider minimally branching partial orders on the repetition roots, two overlapping chains. Together with my collaborator mentioned above, Robert Mercas, we formed a strategy for tackling this next step, using the so called "anchors" introduced for the single chain case. We tried several modifications to the anchor definitions and found promising candidates.
Apart from pursuing the generalization in the distinct repetition case, I will work on extending the approach of counting roots in the general case of runs, in order to better integrate different kinds of repetitions into a unified framework.
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| 次年度使用額が生じた理由 |
The reason for incurring amount to be used next fiscal year was that I had to postpone my visit to the United Kingdom. It was planned for March 2020, to Loughborough University, to work together with my collaborator Robert Mercas, but due to the Covid-19 pandemic, international travel restrictions made it impossible to visit.
I am planning on visiting the UK as soon as the pandemic restrictions are lifted, hopefully in the next fiscal year.
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