A Connection Between Unbordered Partial Words and Sparse Rulers
<|reference_start|>A Connection Between Unbordered Partial Words and Sparse Rulers: $\textit{Partial words}$ are words that contain, in addition to letters, special symbols $\diamondsuit$ called $\textit{holes}$. Two partial words of $a=a_0 \dots a_n$ and $b=b_0 \dots b_n$ are $\textit{compatible}$ if for all $i$, $a_i = b_i$ or at least one of $a_i, b_i$ is a hole. A partial word is $\textit{unbordered}$ if it does not have a nonempty proper prefix and a suffix that are compatible. Otherwise the partial word is $\textit{bordered}$.   A set $R \subseteq \{0, \dots, n\}$ is called a $\textit{complete sparse ruler of length $n$}$ if for all $k \in \{0, \dots, n\}$ there exists $r, s \in R$ such that $k = r - s$. These are also known as $\textit{restricted difference bases}$.   From the definitions it follows that the more holes a partial word has, the more likely it is to be bordered. By introducing a connection between unbordered partial words and sparse rulers, we improve bounds on the maximum number of holes an unbordered partial word can have over alphabets of sizes $4$ or greater. We also provide a counterexample for a previously reported theorem.   We then study a two-dimensional generalization of these results. We adapt methods from one-dimensional case to solve the correct asymptotic for the number of holes an unbordered two-dimensional binary partial word can have. This generalization might invoke further research questions.<|reference_end|>
@article{saarela2024a,
    title={A Connection Between Unbordered Partial Words and Sparse Rulers},
    author={Aleksi Saarela and Aleksi Vanhatalo},
    journal={arXiv preprint arXiv:2408.16335},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.16335},
    primaryClass={math.CO cs.FL}
}