The binary digits of a power

Abstract:

Let $B(m)$ denote the number of ones in the binary expansion of the integer $m \geqslant 1$ and let ${r_h}(m) = B({m^h})/B(m)$ where h is a positive integer. The maximal order of magnitude of ${r_h}(m)$ is $c(h){(\log m)^{(h - 1)/h}}$ where $c(h) > 0$ depends only on h. That this is best possible follows from the Bose-Chowla theorem. The minimal order of magnitude of ${r_2}(m)$ is at most $c{(\log \log m)^2}/\log m$ where $c > 0$ is an absolute constant.