Timeline for Finding limit points of $\{2^n \sqrt 2\}$
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 7, 2023 at 8:16 | comment | added | Pavel Gubkin | Here is the related question on SE: math.stackexchange.com/questions/3573684/…. Not an answer to the question but there are some estimates for number of ones and zeroes in the binary expression of $\sqrt{2}$. | |
| Feb 4, 2023 at 4:21 | comment | added | Anthony Quas | It is certainly known that there is more than one limit point. Write out the base 2 expansion of $\sqrt2$. As @geraldedgar says, Doubling and taking the fractional part is equivalent to shifting the binary expansion left and truncating the first term. Since $\sqrt 2$ is irrational, the binary expansion is not periodic. It follows that the set of limit points is infinite. | |
| Feb 4, 2023 at 1:28 | comment | added | Gerald Edgar | It is suspected (but not proved) that $\sqrt{2}$ is normal in all bases. This would imply that $\{2^n \sqrt{2}\}$ has every element of $[0,1]$ as limit point. The reason is that the binary expansion for $\{2a\}$ is obtained as the left shift of the binary expansion of $\{a\}$. | |
| Feb 4, 2023 at 0:31 | comment | added | tj_ | @Anthony Quas: Your comment indicates that you know the answer. Would you please share your knowledge with the community. Thank you. | |
| Feb 3, 2023 at 22:07 | comment | added | Rabat | It is related to a research question. I was reading a survey on some Ramanujan's diophantine equations (see e.g. pp. 12-14 in "The problems submitted by Ramanujan to the indian mathematical society", and I though that some of them can be related to the posted question! | |
| Feb 3, 2023 at 18:39 | comment | added | Christophe Leuridan | I do not know whether the answer is known. For example, a similar question, the equirepartition modulo 1 of $((3/2)^n)_{n \ge 0}$ is an open question (to my knowledge). Indicating where the questions come from would be useful: is it related to a question in research or does it come from an exercise? | |
| Feb 3, 2023 at 18:24 | history | edited | LSpice | CC BY-SA 4.0 |
TeX in title; deleted "thanks"
|
| Feb 3, 2023 at 17:47 | review | Close votes | |||
| Feb 13, 2023 at 3:08 | |||||
| Feb 3, 2023 at 17:32 | comment | added | Anthony Quas | Is this a homework question? It is not at the appropriate level for mathoverflow. | |
| S Feb 3, 2023 at 17:29 | review | First questions | |||
| Feb 4, 2023 at 2:00 | |||||
| S Feb 3, 2023 at 17:29 | history | asked | Rabat | CC BY-SA 4.0 |