New dimension bounds for $\hat{I}\pm {\hat{I}}^2$ sets

Abstract

In this paper we obtain new lower bounds for the upper box dimension of $\hat{I}\pm {\hat{I}}^2$ sets. As a corollary of our main result, we show that if $\hat{I}\pm$ is not a Liouville number and ${\hat{I}}^2$ is a Liouville number, then the upper box dimension of any $\hat{I}\pm {\hat{I}}^2$ set is $1$. We also use our dimension bounds to obtain new results on affine embeddings of self-similar sets.