A badly expanding set on the $2$-torus

Abstract

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