Rate of appearance of digits 7 and 8 in a given sequence

Question

This is from Arnold's Mathematical Methods of Classical Mechanics: let us consider the sequence $$ 1,\ 2,\ 4,\ 8,\ 1,\ 3,\ 6,\ 1,\ 2,\ 5,\ 1,\ 2\ \ldots $$ which consists of the first digits of the powers of $2$: $\left\{2^n\right\}$ where $n=0,1,2\ldots$

Arnold states that the digit $7$ shows up more often than $8$ in this sequence with a rate of: $$ \frac{\log 8 - \log 7}{\log 9 - \log 8}. $$ My (unsuccessful) attempt:

This "corollary" appears in the chapter regarding the average theorems: temporal and spatial average. In particular it occurs after a corollary stating that, given a Jordan-measureable region $D$ of the $n$-dimensional torus $T^n$, if $\tau_D(T)$ is the quantity of time during which the interval $0\le t\le T$ of the trajectory $\varphi (t)$ is found in $D$, then:

$$ \lim_{T\to \infty}\frac{\tau_D(T)}{T} = \frac{\mu (D)}{(2\pi)^n} $$

The thing is I don't get how the two statements are correlated! Thanks in advance.

Answer 1

The one-dimensional torus here is $[0,1)$, the reals modulo $1$, ans shall represent $\log(2^n)\bmod 1$. The transformation is therefore $x\mapsto x+\log 2\bmod 1$. The region corresponsing to "begins with digit $7$" is therefore $[\log 7,\log 8)$.

Answer 2

Ok perhaps I've figured out something with the help of a friend of mine: let $n\in \mathbb{N}$ be our succession member and $m\in\mathbb{N}$ satisfying $10^m \le n < 10^{m+1}$. We can now express $n$ as $n=10^mx$, where $x\in[1,10)\subset\mathbb{R}$, or in other words: $\log n=\log x$ mod$1$.

Now, $x=a,b$ where $a,b\in\mathbb{N}$ and $a\in \{0,1,2,3,4,5,6,7,8,9\}$ is our desired first digit.

We have: $a\le x < a+1$ by definition and so $a=7$ iff $$ 7\le x<8 \\ \log 7\le \log x< log 8 \\ \log 7\le \log n< log 8\ \ mod1 $$ finally we get the measure of our region by plain subtraction of the two extreme values: $\log8 - \log7$.

Incidentally we think it may work for, say, $3$ or $7$.