Timeline for Distribution of the first occurrence of a maximum (record) run of zeros in the digits of a normal number (say $\pi$)
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Feb 8, 2021 at 3:04 | comment | added | Vincent Granville | Thanks! Looking forward to reading the article you mentioned. | |
| Feb 8, 2021 at 3:00 | comment | added | Yuval Peres | Thanks. Yes, if $b$ is a quadratic irrational the corresponding expansion yields a Markov chain (see e.g. [IT]) and the theory of runs has been extended to that case. [IT] jstage.jst.go.jp/article/jmath1948/26/1/26_1_33/_pdf | |
| Feb 8, 2021 at 2:42 | vote | accept | Vincent Granville | ||
| Feb 8, 2021 at 2:33 | comment | added | Vincent Granville | Yes, original question is for $b$ integer, and mostly $b=2$. No need to answer the case $b$ non-integer, I just thought about it as I played in the past with non-integer bases, and it leads to some exciting stuff: approximation of transcendental numbers by quadratic irrationals, rather than by rationals. | |
| Feb 8, 2021 at 1:59 | comment | added | Yuval Peres | In the original question it seemed that $b$ was an integer. All the description and data corresponded to that. The non-integer case is a different question, also very interesting. But substantial changes/extensions to a question after someone answers reduce the effort people will put in to answer questions in the future. | |
| Feb 8, 2021 at 1:15 | comment | added | Vincent Granville | The fact that I used $b=3$ in my simulations is due to a computer glitch. I use the chaotic dynamical system $x_{n+1}=\{bx_n\}$ to generate digits (the brackets denote the fractional part) and after 45 or so iterations, $x_n=0$ if $b=2$ due to the way computer arithmetic is performed in Perl. It words with $b=3$ (or $b=1.999999$ for that matter), the sequence is still wrong due to round-off errors growing exponentially, but at least because of ergodicity, the asymptotic results remain correct if $b$ is an odd integer. | |
| Feb 8, 2021 at 0:55 | comment | added | Vincent Granville | Thank you. I am currently investigating the case $b=\sqrt{2}$ (yes, the base need not be an integer). In that case, the distribution of 0's and 1's (the digits are binary just like for $b=2$) is not uniform, it's a bit like non-independent Bernoulli trials with $p\neq\frac{1}{2}$. Wondering if all the theory you are citing (thank you!) still holds. I may need such results to get best approximations to irrationals, by numbers such as $(a_n+b_n\sqrt{2})/2^n$ where $a_n, b_n$ are integers. | |
| Feb 7, 2021 at 21:19 | history | answered | Yuval Peres | CC BY-SA 4.0 |