2

Let me start by saying that this post is not likely to be a question. I am writing it because I found a mathematical problem that I found interesting enough to share with others. It seems to me that it is rather difficult and thus I rather not expect an answer. What is it about?

Take any two irrational numbers (e.g., the canonical $e$ and $\pi$) and on both positive semi-diagonals of the coordinate system, instead of the natural numbers, start writing out their consecutive digits. This is how we label the axes. Then let's put a black square on the lattice points when its "digital coordinates" have a sum modulo $2$ equal to $1$, and let's not give it when it is equal to $0$. As you can see, the squares begin to cluster into larger groupings, and within these groupings into larger squares (as the picture below).

enter image description here

The question is:

What is the largest area of a square that can be obtained in this way in a "grid of digits modulo" for two specific irrational numbers?

Of course, I can choose two artificial irrational numbers in such a way as to elevate the area of such squares to infinity (again, picture below), but putting two specific irrational numbers in it makes the whole problem quite interesting, because we have a predetermined "grid of digits modulo" that we can't change, and which doesn't seem to have such simple patterns of distribution of small squares (by the fact that we have irrational numbers).

enter image description here

My intuition pushes me, in the direction of saying that such a maximum field will exist, although I could be wrong (due to the fact that the digits seem to behave randomly in this case, and over time you will come across larger and larger groupings of squares if you just wait long enough. Such a hypothesis. Then one could ask about the expected value of the area of a random square spanning maximally the local group, etc.).

I am simply curious about your impressions and whether problems of this type appeal to you. Regards.

Mat196
  • 33
  • 1
    Your question isn't very clear. Are you asking specifically about $e$ and $\pi$ or what? – Rob Arthan Aug 26 '22 at 21:18
  • If you want you can elaborate on $e$, $\pi$ or any numbers. Whatever you want. Why put artificial limits on yourself? – Mat196 Aug 27 '22 at 07:36
  • 1
    That doesn't make your question any clearer. – Rob Arthan Aug 28 '22 at 00:59
  • If the two irrational numbers are both normal with respect to base ten (or base two as case may be), then it follows that there is no upper bound, because both expansions will contain arbitrarily long sequences of digits of a selected parity. The trouble is that normality of neither $\pi$ nor $e$ is known. IIRC both are suspected to be normal (or was it $\sqrt2$), but we don't know the answer. – Jyrki Lahtonen Aug 28 '22 at 07:44
  • Your construction does not require normality, but it may still be that a all we can say is that with a high probability there is no upper bound to the size of the squares. – Jyrki Lahtonen Aug 28 '22 at 07:45

0 Answers0