$\pi$ and $e$ as coded trajectories

Question

Question about the number $\pi$ and $e$ and their unpredictability.

We know that $\pi=3.141592653589793238462643383279502884...$ Suppose that we are in the origin of the plane i.e. at the point $(0,0)$ and we have possibility to move according to digits of $\pi$.

Let

digit $1$ be code for move $[1,0]$ (vector of translation),
2 for move $[0,1]$,
3 for move $[-1,0]$,
4 for move $[0,-1]$,
others digits of $\pi$ to be ignored.

• Could we somehow prove that starting from the origin and taking the first digit $3$ as the first command of movement, then 1, then 4 etc ..... we can move outside area of a circle with any given radius after enough long time or there is however some boundary for our potential infinite movements ?

• And what about $e=2.71828182845904523536...$, maybe for all transcendental numbers such trajectory, I suspect, is unbounded ?
  (could we say that $e$ is in the same class of difficulty for analysis as $\pi$ or however in some lighter class?)

• $^{Additionally}$ Could trajectories for $\pi$ and $e$ when they were been started at the same time (and it is a single movement for them in the unit time) would meet one day on 2D plane?

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Edit after 5 days

If the above questions are too difficult to tackle couldn't we try to determine at least whether the difference between even and odd digits of $\pi$ or $e$ is unbounded or not.. the numbers for consideration can be also in binary format so the question would be about difference of sums of digits $1$ and digits $0$ in these numbers for approximations of $\pi$ or $e$ with $n$ digits denoted as $\pi_n$ or $e_n$.

Then difference (for $\pi$) can be denoted as $\Delta_{oe}(\pi_n)=s_o(\pi_n)-s_e(\pi_n)$ or for binary version $\Delta_{10}(\pi_n)=s_1(\pi_n)-s_0(\pi_n)$ what is equal $2s_1(\pi_n)-n$.

So if $n$ (number of known binary digits) is increasing with the time of calculations $t$ (we can assume any relation between $n$ and $t$, also linear) the problem is equivalent to determine whether fluctuations $s_1(\pi_n)$ over and under line $f(t)=n/2$ are bounded or not in reference to this line..

Answer 1

I'd say it's unbounded but it'll be hard to prove.

Here is a picture of the trajectory of the first 100000 points and a graph of the distance to the origin for this trajectory. The largest distance is $\sqrt{45520} \approx 213$.

Answer 2

It is extremely unlikely that you're going to be able to get any answer for any question of this sort. Questions about the decimal expansions of irrational numbers are totally unconnected to anything we'd normally consider a good mathematical property of the numbers.

To illustrate how complete our lack of understanding is here, we cannot currently rule out the following statement:

The googol-th digit in the decimal expansion of pi is a 7, and after this point every single digit is either a 7 or a 9. In particular, all the other digits only appear finitely many times in the decimal expansion.

Oh, and here's another statement we also can't currently rule out:

$\pi + e$ is a rational number.

I'm more confident that these statements are false than I am that the sun will come up in the morning, but I'll also not optimistic that either will be disproven in my lifetime. (Note that the former statement, if true, would give answers to most of your questions that are not the expected ones.)