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An isoperimetric inequality for word overlap

Dmitrii Zakharov

Abstract

Let $A$ and $B$ be sets of words of length $n$ over some finite alphabet. Suppose that no suffix of a word in $A$ coincides with a prefix of a word in $B$. Then we show that the product of densities of $A$ and $B$ is upper bounded by $1/n$. This bound is sharp up to a factor of $e$.

An isoperimetric inequality for word overlap

Abstract

Let and be sets of words of length over some finite alphabet. Suppose that no suffix of a word in coincides with a prefix of a word in . Then we show that the product of densities of and is upper bounded by . This bound is sharp up to a factor of .
Paper Structure (2 sections, 2 theorems, 31 equations)

This paper contains 2 sections, 2 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:isopery']} and a corollary

Key Result

Theorem 1.1

We have $\gamma(\alpha, n) \leqslant \frac{(1-\alpha)(1-(1-\alpha)^n)}{\alpha n}$ for all $n\geqslant 1$ and $\alpha \in (0, 1)$.

Theorems & Definitions (4)

  • Theorem 1.1
  • proof
  • Corollary 2.3
  • proof