Lawrence Berkeley National Laboratory

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