Timeline for Representing numbers in a non-integer base with few (but possibly negative) nonzero digits
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
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| Jan 18, 2010 at 15:49 | comment | added | aorq | Thanks for the clarifications. It appears your ideas have inspired Bjorn Poonen to come up with a proof ... | |
| Jan 18, 2010 at 14:29 | history | edited | Douglas Zare | CC BY-SA 2.5 |
Deleted incorrect proof.
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| Jan 18, 2010 at 4:32 | comment | added | Douglas Zare | If $b$ is a root of a polynomial, then you can find polynomials of $b$ of arbitrarily large degree and fixed total coefficient magnitude with small values. For example, suppose you are trying to express the integers up to 10. $\phi^{1000}-\phi^{999}-\phi^{998}+\phi^4 < 10$, and is of high degree. However, it's not a new value. The higher powers are removable, so that you find a lower degree polynomial $\phi^4$ with the same value. It looks like I should clean up the proof of that lemma ("fixed" not "finite"), but you need something like that lemma to use a simple counting argument. | |
| Jan 17, 2010 at 17:08 | comment | added | aorq | I'm sorry, but I don't understand your proof. Where does $log_b(k-1)$ come from? How do you get that $b$ is a root of some limit polynomial? What does it matter if $b$ is a root of a polynomial "of finite degree"? | |
| Jan 17, 2010 at 13:42 | history | answered | Douglas Zare | CC BY-SA 2.5 |