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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.

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

Wikipedia even has a section of its article about normal numbers stating exactly this, and giving credit to a book from 2003 and a paper by Bugeaud from 2012.

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  • $\begingroup$ Thank you, and this is good news. I knew the theorem in question, but could not figure out that it was equivalent to being a good seed. $\endgroup$ – Vincent Granville Sep 4 at 15:28

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