Every strongly summable ultrafilter on $\bigoplus\mathbb Z_2$ is sparse

TL;DR

This paper proves that all strongly summable ultrafilters on the countably infinite Boolean group are sparse, resolving a question about their existence and properties in this specific algebraic setting.

Contribution

It demonstrates that every strongly summable ultrafilter on the countably infinite Boolean group is sparse, providing a definitive answer to a previously open question.

Findings

All strongly summable ultrafilters on the Boolean group are sparse

Addresses a question posed by Hindman, Steprns, and Strauss

Contributes to understanding ultrafilter structure on abelian groups

Abstract

We investigate the possibility of the existence of nonsparse strongly summable ultrafilters on certain abelian groups. In particular, we show that every strongly summable ultrafilter on the countably infinite Boolean group is sparse. This answers a question of Hindman, Stepr\=ans and Strauss.