2022 Fiscal Year Research-status Report
Clusters of repetition roots
| Project/Area Number |
19K11815
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| Research Institution | Akita University |
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Principal Investigator |
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| Project Period (FY) |
2019-04-01 – 2024-03-31
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| Keywords | combinatorics on words / repetitions / squares / distinct squares / upper bounds |
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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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| Causes of Carryover |
Due to the pandemic some of the in-person meetings with main collaborator Robert Mercas from Loughborough university has been delayed, hence the remaining funds. I am planning to use the remaining amount for conference/publication expenses when submitting the final work proving the general case of our conjecture.
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