# Enhanced Berry-Esseen theorem for the digits of $\sqrt{2}$

The Berry-Esseen theorem provides a second-order approximation to the central limit theorem (itself a first order approximation.) Higher order approximations are available, see here. If the random variables are Bernouilli($$p$$), better approximations are available, this is a topic of active research. Before stating my question, Let me mention a recent result, which is the topic of a PhD thesis (the full thesis available here):

It seems that my question (below) is just a particular case of this result, not sure. Probably it is, but maybe I got it wrong.

Question

The proportion $$q_n$$ of binary digits of $$\sqrt{2}$$ (or any other normal number for that matter, say $$\pi$$ or $$\ln 2$$), among the first $$n$$ digits, satisfies

$$\sqrt{n}\cdot\Big|q_n-\frac{1}{2}\Big|<\frac{1}{\sqrt{2\pi}}\approx 0.398942 \mbox{ as } n\rightarrow\infty.$$

This assumes $$p=\frac{1}{2}$$. Below is the chart showing $$\sqrt{n}\cdot|q_n-\frac{1}{2}|$$ up to $$n = 1,000,000$$.

I don't expect an answer for $$\sqrt{2}$$ nor for any other normal number. All I want to known is, if the digits were i.i.d Bernouilli($$\frac{1}{2}$$), would this be correct? See related questions on Math.StackExchange, here and here.

The goal is to try to prove a much weaker result for $$\sqrt{2}$$ hoping it would be easier than proving the result stated in my question, maybe something like $$\log \log n \cdot |q_n - \frac{1}{2}| = o(1)$$. That would be enough to prove that the binary digits of $$\sqrt{2}$$ are zero 50% of the time.

• Possibly mathoverflow is better for you – Eduardo S. Aug 26 '19 at 17:15
• Thanks, I am thinking posting on crossvalidated too. While my interest is purely number theory, this question is about probability theory and could prove useful in many other, non-mathematical contexts. – Vincent Granville Aug 26 '19 at 17:17
• @VincentGranville It might be worth mentioning that we do not know whether $\pi$ , $e$ or any irrational algebraic number is normal in base $2$. Perhaps, more is known about the frquency of binary digits in $\sqrt{2}$. – Peter Aug 27 '19 at 10:28
• @Peter: Exactly, and people should NOT try to prove this fact for these numbers (I once offered a \$500k award for this!), but instead for random sequences. – Vincent Granville Aug 27 '19 at 12:58
• – Peter Aug 27 '19 at 13:29