A badly expanding set on the $2$-torus

TL;DR

This paper provides a counterexample to a conjecture about expansion properties on the 2-torus under specific linear transformations, challenging previous assumptions in the field.

Contribution

The authors construct a counterexample to a longstanding conjecture about expansion on the 2-torus under certain matrices.

Findings

Counterexample disproves the conjecture

Expansion properties on the 2-torus are more nuanced than previously thought

Challenges existing theories on linear transformations and expansion

Abstract

We give a counterexample to a conjecture stated in Linial and London 2006 regarding expansion on $\mathbb{T}^2$ under $\begin{bmatrix}1&1\\ 0&1\end{bmatrix}$ and $\begin{bmatrix}1&0\\ 1&1\end{bmatrix}$.