Thick Difference Sets of Haar Null Compact Sets in Locally Compact Groups

Let $G$ be a non-discrete, locally compact group with Haar measure $m$. We prove that there exists a compact set $K\subset G$ with $m(K)=0$ such that $K{K}^{-1}$ contains a neighborhood of the identity. Moreover, such a set may be constructed inside any prescribed neighborhood of the identity.