  In this paper we obtain new lower bounds for the upper box dimension of
$\alpha\beta$ sets. As a corollary of our main result, we show that if $\alpha$
is not a Liouville number and $\beta$ is a Liouville number, then the upper box
dimension of any $\alpha\beta$ set is $1$. We also use our dimension bounds to
obtain new results on affine embeddings of self-similar sets.
