The Mysterious World of Normal Numbers

摘要

Given an integer q ≥ 2, a q-normal number (or a normal number) is a real number whose q-ary expansion is such that any pre-assigned sequence of length k ≥ 1, of base q digits from this expansion, occurs at the expected frequency, namely 1/q~k. Even though there are no standard methods to establish if a given number is normal or not, it is known since 1909 that almost all real numbers are normal in every base q. This is one of the many reasons why the study of normal numbers has fascinated mathematicians for the past century. We present here a brief survey of some of the important results concerning normal numbers.