A strong hot spot theorem

A real number $\alpha$ is said to be $b$-normal if every $m$-long string of digits appears in the base-$b$ expansion of $\alpha$ with limiting frequency $b^{-m}$. We prove that $\alpha$ is $b$-normal if and only if it possesses no base-$b$ ``hot spot''. In other words, $\alpha$ is $b$-normal if and only if there is no real number $y$ such that smaller and smaller neighborhoods of $y$ are visited by the successive shifts of the base-$b$ expansion of $\alpha$ with larger and larger frequencies, relative to the lengths of these neighborhoods.