A strong hot spot theorem

Abstract

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