Upper bound for the number of privileged words

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21340%2F23%3A00361404" target="_blank" >RIV/68407700:21340/23:00361404 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1016/j.disc.2022.113164" target="_blank" >https://doi.org/10.1016/j.disc.2022.113164</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1016/j.disc.2022.113164" target="_blank" >10.1016/j.disc.2022.113164</a>

Result language
angličtina
Original language name
Upper bound for the number of privileged words
Original language description
A non-empty word w is a border of a word u if |w| < |u| and w is both a prefix and a suffix of u. A word u is privileged if |u| <= 1 or if u has a privileged border w that appears exactly twice in u. Peltomaki (2016) presented the following open problem: "Give a nontrivial upper bound for B(n)", where B(n) denotes the number of privileged words of length n. Let ln([0])(n) = n and let ln([j])(n) = ln(ln([j-1])(n)), where j, n are positive integers. We show that if q > 1 is a size of the alphabet and j >= 3 is an integer then there are constants a_j and n_j such that B(n) <= a_j q(n)root(ln n)/root(n)*In[j](n) Product_(i=2)^(j-1) root(ln[i](n)), where n >= n_j. This result improves the upper bound of Rukavicka (2020).
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Publication year
2023
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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