Clusters of repetition roots
Project/Area Number |
19K11815
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Research Category |
Grant-in-Aid for Scientific Research (C)
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Allocation Type | Multi-year Fund |
Section | 一般 |
Review Section |
Basic Section 60010:Theory of informatics-related
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Research Institution | Akita University |
Principal Investigator |
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Project Period (FY) |
2019-04-01 – 2024-03-31
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Project Status |
Granted (Fiscal Year 2022)
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Budget Amount *help |
¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000)
Fiscal Year 2021: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2020: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2019: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
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Keywords | combinatorics on words / repetitions / squares / distinct squares / upper bounds / square network / distinguished positions / Repetitions / Squares / Combinatorics on words / Distinct repetitions |
Outline of Research at the Start |
The main questions in this research project are: how many repetition occurrences and how many unique repetition types (distinct repetitions) can there be in a word (sequence)? I aim to improve existing bounds for distinct repetitions, in particular, tackling a famous conjecture by Fraenkel and Simpson on the number of squares. I propose a fresh approach to understand the structure of distinct repetitions through clusters of their roots, which is expected to lead to improvements in the bounds and provide an easy, visually informative way of presenting their proofs.
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Outline of Annual Research Achievements |
In the last year we managed to extend the technique of anchor positions to represent repetitions with exponents higher than 2. Using this, we showed that our cluster size conjecture holds for single chains of repetition roots with arbitrary integer exponent. We also showed that the bounds given for single chains are optimal by a constructive proof yielding sequences of clusters for any combination of cluster sizes allowed within the bounds. These results have been published in DCFS 2022. A big development from last year was the solution of the Fraenkel-Simpson conjecture by Li and Brlek using properties of Rauzy graphs. We tried to extend their arguments to so called extended Rauzy graphs to prove our clusters conjecture in the general case. The paper presenting those results is under work as of now.
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Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
Due to the recent proof of the square conjecture by Li and Brlek, we changed strategy but the goal remains the same. Their Rauzy graph method is very powerful and it looks like we can employ it to prove our more general conjecture more easily than using the anchors directly.
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Strategy for Future Research Activity |
At the moment we are working on the details of extending the Rauzy graph concept and adapting Li and Brlek's method to that more general setting to finish the proof of the clusters conjecture in the general case. We will look into possible small improvements of the bound away from n. We will also investigate the optimality of the bounds given by the conjecture in the sense mentioned earlier for single chains: for arbitrary cluster sizes allowed within the bounds try to find a sequence of repetitions whose roots form clusters of those given sizes.
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Report
(4 results)
Research Products
(14 results)