I will try to give a different point of view on this question. It will be more geometrical and not exactly answering the question but that might still give some insight.
I will talk about Constructible numbers
. The principle is the following. You start with the two points at distance 1 on a plane, and you have a compass and a straightedge, and you try to construct new points from existing one by taking the intersection points of lines going through any of the two points and circles centered at one point with radius equal to a distance between two existing points.
For convenience I will see the points in the plane as complex numbers. A number is constructible if it can be obtained by this process.
Let’s give the first examples :
From the first two points 0 and 1, I can draw the line passing though these two points, the two circles of radius 1 centered on 0 and 1 respectively. This gives me 4 new points : -1, 2 , eipi/6 , e-ipi/6
From this new set of six points, I can draw 10 different lines. And I have four possible radius for my circles (1,2,3,sqrt(3)) and 6 possible centers which gives me more or less 24 new circles. So I get more than 100 new points (see the picture
And so on … the number of points explodes very quickly.
Ok what was the point of all this ? Well it is interesting to note that all the (non-zero) numbers we can construct in 2 steps all have norm between 0.2 and 5. And more importantly, most (like 95% may be, I didn’t compute it) have norm between 0.5 and 4. (see the picture [RES ignored duplicate image]
When I go on with the steps I see that there is always a lower bound and an upper bound (obvious), and that most of the numbers have norm between 0.5 and 4. (less obvious but it’s because, when you draw all the circles around your existing points, most of them will stay in the same range …).
Now that we understand that, what is the point of all this ? Well, suppose we could see the property of being “interesting” as a random occurence among numbers (this is my hypothesis, I am not exactly proving that this means anything). And to construct interesting numbers we have to do a certain number of steps. Equivalently, I construct all possible points in n steps and pick randomly some of them (to say that they are interesting). With the arguments above, this would prove that :
a majority (I have no idea how to exactly compute it) of the interesting numbers constructible in n steps lie in a rather restricted range between 0.5 and 5.
a few will be larger or smaller than that …
And no interesting number constructible in n steps will have norm smaller than 10-n or be bigger than 10n(because no number constructible in n steps can be that big)
Tl; DR : If I randomly pick numbers that could be defined in a certain number of steps, then most of the numbers I picked will be in the [0.5 , 5] range
Now, back to mathematical constant. How are they defined ? Well, we have to do several steps. Much more complicated steps than for constructible numbers, but still a certain number of steps. (I could look at the numbers that can be defined in a certain number of words, to get an idea, even if this definition is problematic).
So I would guess (and this is wishful thinking) that the principle stay the same. Of all the numbers that can be defined in a certain number of steps, most of them should in the zone around 0.5 and 5. So the same should be true for interesting numbers …
Thank you for your attention. I hope this was not too confusing.