Almost solutions of equations in permutations

TL;DR

This paper investigates conditions under which approximate solutions to permutation equations can be closely matched by exact solutions, providing criteria and examples of when such solutions do or do not exist.

Contribution

It introduces sufficient conditions for the existence of exact solutions near approximate solutions in permutation equations and presents examples illustrating non-existence cases.

Findings

Sufficient conditions for the existence of exact solutions near approximate solutions.

Examples demonstrating cases where no exact solution exists near an approximate one.

Abstract

We will say that the permutations f_1,...,f_n is an e-solution of an equation if the normalized Hamming distance between its l.h.p. and r.h.p. is less than e. We give a sufficient conditions when near to an e-solution exists an exact solution and some examples when there does not exist such a solution.