This is an announcement for the paper "On the Hardy-Littlewood maximal function for the cube" by Jean Bourgain. Abstract: It is shown that the Hardy-Littlewood maximal function associated to the cube in $\mathbb R^n$ obeys dimensional free bounds in $L^p$ fir $p>1$. Earlier work only covered the range $p>\frac 32$. Archive classification: math.FA Mathematics Subject Classification: 42B25 Remarks: 20 pages Submitted from: bourgain@ias.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1212.2661 or http://arXiv.org/abs/1212.2661