Rado equations solved by linear combinations of idempotent ultrafilters

TL;DR

This paper characterizes when Rado equations can be solved within linear combinations of idempotent ultrafilters, expanding previous partial results by leveraging relations between ultrafilter combinations and integer sequences.

Contribution

It provides a complete characterization of the solvability of Rado equations in linear combinations of idempotent ultrafilters, generalizing earlier partial results.

Findings

Full characterization of Rado equations solvability in ultrafilter combinations

Connection established between ultrafilter combinations and integer sequences

Generalization of previous partial results by Mauro Di Nasso

Abstract

We fully characterise the solvability of Rado equations inside linear combinations $a_{1}\U\oplus\dots\oplus a_{n}\U$ of idempotent ultrafilters $\U\in\beta\Z$ by exploiting known relations between such combinations and strings of integers. This generalises a partial characterization previously obtained by Mauro Di Nasso.