Uniform distribution Theory, Spring 2021

Hint for Problem 1 in assignment 2

The question asked to show that simple normality in base b^k implies normality. This is NOT the same as showing that the occurence of every string of length say 2 is equivalent to simple normality in base b² (which is not true).

What you have to show is that simple normality in base b^k for ALL k imples that the frequencty of appearances of say 00 is 1/ b².

Each k will give a lower bound (which depends on k), which is a computation that you need to do. Taking the limit will give a lower bound of 1/ b².

Since this is true for every string of length 2, it is also an upper bound.