Labelling n-points in [0,1] arbitrarily from {0,1,..,a-1} I was reading a paper and found this result which was given without proof and couldn’t prove it. “Choosing n points in [0,1] and labelling them with their leading digits and applying map $x\to ax(mod 1)$ ( i.e fractional part of $ax$)“ is the same as “choosing n points in [0,1] and labelling them arbitrarily with integers from {0,1,2,3,..,a-1}” I am a bit confused as to what the statements mean. Can someone explain and give me a hint?

Well this is clear if the statement “leading digits” refers to the base-$a$ representation of $x$ and not the decimal representation of $x$. With this understanding, the statement is just saying that the leading digit has the same probability law as the next to the leading digit, which is trivial. Going back to mentioned article reveals that this is indeed the case here.