  Let $H_k$ be the one dimensional Hilbert transform computed in the direction
$(1,2^k)$ in the plane. We show that the maximal operator $\sup_k |H_kf|$ maps
$L^p$ of the plane into itself for $1<p<\infty$.
  The same result with the Hilbert transform replaced by the one dimensional
maximal function was proved by Nagel, Stein and Wainger in 1978.
