Timeline for Lower bounds for pattern complexity of aperiodic subshifts
Current License: CC BY-SA 4.0
9 events
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| Nov 11, 2022 at 13:12 | comment | added | Ronnie Pavlov | Certainly not for literally aperiodic configurations, since the "skew lines" example I gave in 3 dimensions is on the order of $n^2$, not $n^3$. I personally think there is some version of aperiodicity which does imply what you wrote, but the only such result I know is what Ville mentioned; the Kari-Szabados paper proves a sense in which $c_{n} \leq n^d$ for any $n$ implies that all $x \in X$ are the sum of finitely many periodic configurations. The snag is that they're possibly on infinite alphabet(s). But that means "not sum of periodic configs on inf alphabets" implies your liminf = 1. | |
| Nov 11, 2022 at 8:51 | comment | added | Keen-ameteur | Thank you for your answer. I was not sure about the strict inequality I wrote, but the motivation to my question was the estimates by Cyr and Kra. I was interested in whether we can say that $\liminf \frac{c_n(\Omega)}{n^d}>0$? | |
| Nov 11, 2022 at 8:07 | comment | added | Ville Salo | (I upvoted your comment mentioning me so that it doesn't look like I've upvoted your comment mentioning me.) | |
| Nov 11, 2022 at 4:45 | comment | added | Ville Salo | Heh. Note that Kari-Szabados also gives you another sense in which the two-lines example is periodic: Every configuration that satisfies the Nivat assumption is a finite sum of periodic configurations (in general they need to be over infinite alphabets, but in the case of two lines they are not). | |
| Nov 10, 2022 at 19:42 | history | edited | Ronnie Pavlov | CC BY-SA 4.0 |
Fixed errors noted by Ville Salo in comments
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| Nov 10, 2022 at 19:37 | comment | added | Ronnie Pavlov | I was virtually positive when writing this that (a) I would make mistakes and (b) that you specifically would catch them :) Thanks, and I will go back and update the answer. | |
| Nov 10, 2022 at 16:39 | comment | added | Ville Salo | Also I think it's not true that Nivat has been shown for algebraic subshifts. Rather it's known that the Nivat assumption implies that the configuration belongs to an algebraic subshift, and then we know that some algebraic subshifts cannot appear. | |
| Nov 10, 2022 at 16:30 | comment | added | Ville Salo | Doesn't the Kari-Szabados result sayThe existence of an aperiodic point implies that $p_{m,n}(X) \geq mn+1$ for some (even almost all) $m,n$, contrary to the claim in your first paragraph? | |
| Nov 10, 2022 at 16:22 | history | answered | Ronnie Pavlov | CC BY-SA 4.0 |