On the number of gapped repeats with arbitrary gap

TL;DR

This paper investigates the maximum number of $f,g$-gapped repeats in words of fixed length, establishing linear bounds under weak conditions on the functions defining the gaps.

Contribution

It introduces a general framework for analyzing $f,g$-gapped repeats and derives linear upper bounds for their number in words of length $n$.

Findings

Linear upper bounds for the number of $f,g$-gapped repeats

Applicable to a wide class of functions $f(x)$ and $g(x)$

Extends previous results on repeats with fixed or simple gap constraints

Abstract

For any functions $f(x)$, $g(x)$ from $\mathbb {N}$ to $\mathbb {R}$ we call repeats $uvu$ such that $g(|u|)\le |v|\le f(|u|)$ as {\it $f,g$-gapped repeats}. We study the possible number of $f,g$-gapped repeats in words of fixed length~$n$. For quite weak conditions on $f(x)$, $g(x)$ we obtain an upper bound on this number which is linear to~$n$.