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I understand that if some number $x$ is a normal number, its digits follow a uniform distribution in any base. Additionally, I understand that any particular sequence of $k$ digits of $x$ in base $b$ will occur with density $b^{-k}$

I now wonder whether there must exist some number $n$ for which the sequence digits of $x$ indexed by 1 through $n$ is identical to that indexed by $n+1$ through $2n$.

By this definition, the likelihood of finding the first $n$ digits repeated beginning at digit $n+1$ is $b^{-n}$. Then, the likelihood of there existing such a number $n$ in any of the infinitely many digits of $x$ is

$$P(b) = 1-\prod_{n=1}^\infty (1-b^{-n}).$$

This can alternately be expressed as

$$P(b) = 1 - \left(\frac{1}{b}; \frac{1}{b}\right)_\infty,$$

where $(a;q)_N$ is the q-Pochhammer symbol . I can approximate this expression numerically without much trouble, and the results for a few values of $b$ are below.

$$ \begin{array}{c|c} b & P(b) \\ \hline 2 & 0.71121 \\ \hline 5 & 0.23967 \\ \hline 10 & 0.10999 \\ \hline 16 & 0.066405 \\ \end{array}$$

Clearly, none of these numbers are one. If you're just here for the answer, it's no (unless my reasoning is flawed, and it might be). In bases other than two, most normal numbers do not repeat in this way.

My question: why? What intuition or other manipulation could I apply here? Doing it numerically is pretty inelegant.

In base ten, this reduces to proving that the product $$ (0.9)(0.99)(0.999)... > 0. $$ (Well, I suppose the above is true of any base if we let 9 be the symbol for the largest valued digit.) This is a pretty simple looking product, but I'm a little stuck just demonstrating that it's nonzero. I can rather abstractly believe that the terms in this product approach one faster than the product approaches zero, but that's really hand wavey. The Laurent polynomial expansion is pretty irregular. Maybe I'm missing something obvious; any thoughts would be appreciated. Bonus points if you can show whether $P(b)$ is less than or greater than one half given some $b$.

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    $\begingroup$ As you sure this property is true? The Champernowne Constant is normal, and doesn't appear to have the given property (although I'm unsure of a proof it doesn't have this property). $\endgroup$ – Mark May 13 '18 at 21:54
  • $\begingroup$ @Mark, looks like the pages for "Normal Number" on both Wikipedia and MathWorld include this property as part of the definition. I imagine that for the Champernowne Constant, it just takes many digits before the statistics flatten out. $\endgroup$ – Anthony May 13 '18 at 22:18
  • $\begingroup$ For any number in which the first $2n$ digits repeat in this way there are $9$ choices of the $2n^{th}$ digit which give a non-repetition. And that is apart from changing the other digits. There are normal numbers that repeat (changing a finite number of digits of a normal number does not affect its normality). Most normal numbers will not repeat, though the sequence of the first $n$ digits will be found infinitely often. $\endgroup$ – Mark Bennet May 13 '18 at 22:22
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$$\prod_{i\ge 1} \left(1-10^{-i}\right)=1-\sum_{i\ge 1}10^{-i}+\sum_{i>j\ge 1}10^{-i-j}-\cdots ~~~~(\star)$$

Now, note that $\sum_{i\ge 1}10^{-i} = \frac{1}{10-1}=\frac{1}{9}<1$.

Hence, $\sum_{i>j\ge 1}10^{-i-j} \le \sum_{j\ge 1}10^{-i}\sum_{i\ge 1}10^{-i}=\frac{1}{9}\sum_{i\ge 1}10^{-i} $. Similarly, each summand in $(\star)$ is smaller in absolute value than the previous term. Hence we have an alternating series with terms monotonically approaching $0$.

In particular, that means $$1>\prod_{i\ge 1} \left(1-10^{-i}\right)>1-\sum_{i\ge 1}10^{-i}=1-\frac{1}{9}=\frac{8}{9}$$

All the above easily generalizes if the base isn't $10$.

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