# About another potential characterization of normal numbers

Normal numbers, in a nutshell, are real numbers that have a "uniform" distribution of digits in standard numeration systems (binary, decimal, and so on.) You can find a formal definition and characterization on Wikipedia. Numbers such as $$e, \pi, \log 2, \sqrt{2}, \gamma$$ are believed to be normal, though there is no proof yet.

A few years ago, I introduced the concept of good seed, which is a different way to characterize "uniformity" or "good behavior" of the digits of a real number $$x$$, called the seed. In short, a good seed is a number whose some associated distribution is the main solution to some specific stochastic integral equation. In dynamical system lingo, that distribution is the main attractor of a particular mapping.

More specifically, $$x \in [0, 1]$$ is a good seed in base $$b$$ if the underlying equilibrium distribution of the ergodic sequence $$z_n = bz_{n-1} - \lfloor bz_{n-1}\rfloor = b^n z_0 - \lfloor b^n z_0\lfloor$$ is uniform on $$[0, 1]$$. Here $$b>1$$ (not necessarily an integer) and $$z_0 = x$$. Also the $$n$$-th digit of $$x$$ in base $$b$$ is $$d_n =\lfloor b z_n\rfloor$$. This is discussed in details in a number of articles that I have written (even a book), one of the most useful ones can be found here even thought it deals mostly with $$1 < b < 2$$.

Among other benefits, the concept of good seed allows you to explicitly compute auto-correlations of lag $$k$$ between successive digits or within the sequence $$z_n$$. For the digits, these auto-correlations are $$0$$ (if $$x$$ is a good seed and $$b$$ is an integer) and for $$z_n$$, the lag-$$k$$ auto-correlation is $$b^{-k}$$ under the same conditions. If $$b$$ is not an integer, the equilibrium distribution is known explicitly only in a few cases, for example for the golden ratio base. It is known to NOT be uniform thus the concept of normal number in (say) base $$5/3$$ is meaningless, while the concept of good seed in base $$5/3$$ makes perfect sense.

Question

This is not about discussing whether my concept of good seed is better or not than the concept of normal number, but rather, if the base $$b$$ is an integer, are these two concepts identical? Maybe testing this on the number such as $$x=0.123456789101112131415...$$ (a normal number in base $$10$$) is good starting point. Or is it almost identical, with only minor differences? In the end, if the two concepts are similar but not identical, it might not matter that much. I am trying to prove something about the digits of $$\sqrt{2}$$ in base $$2$$ that is weaker than both normalcy or being a good seed anyway.

More importantly, I made many claims over the years that the set of numbers that are not a good seed has Lebesgue measure zero. I never proved it, my claim was never questioned, and I borrowed this idea from the fact that this is true for normal numbers. Is my claim correct? Is a good seed a normal number and conversely if the base $$b$$ is an integer? Are the differences minor, if any? These are my main questions.

Yes, for integer $$b$$ it's a reformulation of, and exactly the same as, normality to base $$b$$.