Imperfect friezes of integers

TL;DR

This paper proves that dense subsets of integers contain scaled copies of any finite set, almost perfectly reproducing it for sufficiently large sizes, revealing a form of structural regularity in such sets.

Contribution

It introduces a new result showing that dense integer subsets contain almost perfect scaled reproductions of any finite set for large sizes.

Findings

Dense subsets of integers contain scaled copies of any finite set.

Almost perfect reproductions occur for all sufficiently large sizes.

The result applies to any positive forward density subset of integers.

Abstract

We show that for any positive forward density subset N \subset Z, there exists an integer m>0, such that, for all n>m, N contains almost perfect n-scaled reproductions of any previously chosen finite set of integers.