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Improved Bounds on the Maximum Number of Distinct Squares in Circular Words

Authors Panagiotis Charalampopoulos , Manal Mohamed , Jakub Radoszewski , Wojciech Rytter , Tomasz Waleń , Wiktor Zuba

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LIPIcs.CPM.2026.6.pdf
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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics on words
Keywords
  • circular words
  • squares
  • repetitions

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Abstract

We investigate the asymptotic growth of function CS(n), which maps n to the maximum number of distinct squares in a circular word of length n (that is, the maximum number of distinct squares of length at most n in a word ww of length 2n). We improve upon the lower bound of 1.25n established by Amit and Gawrychowski [SPIRE 2017] and the straightforward upper bound of 2n, which follows from the recent result of Brlek and Li [Comb. Theory, 2025] stating that there are fewer than n squares in standard (i.e., non-circular) words of length n. (Previously, Amit and Gawrychowski gave an upper bound of 32/15n using a weaker upper bound on squares in standard words.) Specifically, we show that CS(n) ≤ ⌈1.8 n⌉ and that, for infinitely many n, CS(n) ≥ 1.5n-𝒪(√n).
For the lower bound, we exploit the combinatorial structure of Fibonacci words to construct a family of square-rich circular words. For the upper bound, we exploit density properties of the starting positions of long squares, adapting an approach of Amit and Gawrychowski.

Cite As Get BibTex

Panagiotis Charalampopoulos, Manal Mohamed, Jakub Radoszewski, Wojciech Rytter, Tomasz Waleń, and Wiktor Zuba. Improved Bounds on the Maximum Number of Distinct Squares in Circular Words. In 37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 369, pp. 6:1-6:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CPM.2026.6

Author Details

Panagiotis Charalampopoulos
  • King’s College London, UK
Manal Mohamed
  • King’s College London, UK
Jakub Radoszewski
  • University of Warsaw, Poland
Wojciech Rytter
  • University of Warsaw, Poland
Tomasz Waleń
  • University of Warsaw, Poland
Wiktor Zuba
  • University of Warsaw, Poland

Funding

  • Radoszewski, Jakub: Supported by the Polish National Science Center, grant no. 2022/46/E/ ST6/00463.

References

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