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September 28

Cohen equivalent of "normal number"

Somewhat vague thoughts I've had for a while, which came to the forefront of my mind based on a recent edit at normal number.

Background: Two of the most basic notions of forcing that everyone needs to know are the OG Cohen forcing, and so-called random real forcing. They have a lot in common. If a real number x is generic for Cohen forcing (over some inner model M, say), that means that for every open dense set U of reals that has a code in the inner model, x is an element of U. If it's generic for random real forcing, it means that for every Borel set A of reals having a code in the inner model, if A is conull (that is, its complement has Lebesgue measure zero), then x is an element of A. (We can pare down from full Borel sets to Gδ sets and get the same notion; that makes it easier to think about the codes.)

Now, it's really really easy to see that a random real over M, for any reasonable M, is normal, but that a Cohen real over M cannot possibly be. This is a good way to think about the divergence between measure and category. For example, if you express a random real in base 17 and look at the fraction of digits that are equal to 13, as the initial segment of the representation gets longer and longer, that fraction has to converge to ${\frac {1}{17}}$. But for the Cohen real it behaves in a much odder way: For any number r between 0 and 1 and any positive ε, there are arbitrarily long initial segments of the representation where the fraction is within ε of r.

My (vague) question is, is this the "right" analogue of normality for Cohen reals? Or is there something stronger we should say, but that's still satisfiable by an actual real number rather than one in a generic extension? --Trovatore (talk) 21:24, 28 September 2026 (UTC)

I thought of a slightly streamlined way of putting it, which it seems to me bolsters the case that it's the "right" analogue, but still interested in hearing thoughts.

Suppose we define, for x a real number, b a natural-number base at least 2, and d a digit from 0 to b−1 inclusive,

$\#(x,b,n,d)=$ the number of occurrences of d in the first n digits of the base-b expansion of x

Now say that x is simply normal to base b if, for every digit d,

$\lim _{n\to \infty }{\frac {\#(x,b,n,d)}{n}}={\frac {1}{b}}$

And say x is normal if it is simply normal to every base. I believe this is the standard definition.

Analogously, say x is simply cat-normal ("category normal") to base b if, for every digit d,

$\limsup _{n\to \infty }{\frac {\#(x,b,n,d)}{n}}=1$

And x is cat-normal if it is simply cat-normal to every base.

Exercises for the reader:

1. This definition of cat-normal is equivalent to the criterion I gave above.
2. Cat-normal reals exist (in fact they're comeager in the reals).
3. Any real that is Cohen over a transitive inner model of ZFC is cat-normal.

Does this seem right, and is it convincing that this is the "right" analogue of normality for Cohen reals? --Trovatore (talk) 02:14, 29 September 2026 (UTC)

Yes, that looks like the right notion. It captures what can be said about the frequency of digits in a Cohen generic. Antendren (talk) 03:41, 29 September 2026 (UTC)

Update: I put the question to Claude, and it pointed me to the concept of "extremely non-normal numbers", which means that for any possible collection of probabilities for digits, there's a subsequence of the initial segments of the representation where the frequencies converge to those probabilities. See here.

At first Claude thought my criterion was weaker than this, but its reason seemed to be considering it base-by-base. I pushed back, saying that my criterion gets stronger than you might expect when you require it for all bases (because you can take lots of digits at a time by looking at, say, base $b^{k}$ for some large $k$). Then it reversed course and said my criterion "for all bases" is equivalent to extremely non-normal in all bases, which I think is probably correct though I haven't tried to completely nail it down.

Aside: This is a little depressing. It's so bloody fast. If math isn't going to be done by humans anymore I wonder if I'm going to like it. It is what it is and there's no point crying about it. But it's my passion and I'll cry if I want to. --Trovatore (talk) 21:34, 29 September 2026 (UTC)