Notes on Gompf's infinite order corks

For any positive integer $n$ we give a ${\mathbb Z}^n$-cork with a ${\mathbb Z}^n$-effective embedding in a 4-manifold being homeomorphic to $E(n)$. This means that a cork gives a subset ${\mathbb Z}^n$ in the differential structures on $E(n)$. Further, we describe handle decompositions of the twisted doubles (homotopy $S^4$) of Gompf's infinite order corks and show that they are Gluck twists and log transforms of $S^4$.