Timeline for Possible limits of $(1/n) \sum_{k=0}^{n-1} e^{i2\pi \cdot 2^k\alpha}$
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11 events
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| Feb 20, 2016 at 18:04 | comment | added | Ian Morris | On the other hand if you were instead to ask about $3^k$ in place of $2^k$, then while it would correspondingly still be true that $L$ is the closure of the barycentres associated to periodic orbits of $x \mapsto 3x \mod 1$, I believe that it is currently unknown which measures correspond to boundary points. | |
| Feb 20, 2016 at 18:03 | comment | added | Ian Morris | As Terence Tao notes, the set of all limits for rational $\alpha$ is a dense subset of $L$. From an abstract perspective, this is due to the fact that invariant measures supported on periodic orbits are weak-* dense in the simplex of invariant measures. The fact that the averages over Bernoulli measures lie in a smaller shape reflects the fact that the closure of this set of measures is much smaller. As is proved in Bousch's paper mentioned in Anthony's answer, the boundary points arise from numbers $\alpha$ whose binary expansion is a Sturmian sequence. | |
| Feb 20, 2016 at 14:34 | vote | accept | Sean Eberhard | ||
| Feb 19, 2016 at 18:18 | answer | added | Anthony Quas | timeline score: 12 | |
| Feb 19, 2016 at 17:57 | comment | added | Sean Eberhard | @BenoîtKloeckner Thanks, I will try googling your suggested buzz words. | |
| Feb 19, 2016 at 17:56 | comment | added | Sean Eberhard | @TerryTao Good point. I think maybe I believe your meta-hand-waving, so that's another useful description of $L$. | |
| Feb 19, 2016 at 17:24 | comment | added | Terry Tao | A trivial remark, improving upon 1: $L$ is contained in the convex hull of the image of $f_n$ for any $n$ (just cut up a long sum $\frac{1}{N} \sum_{k=0}^{N-1} e(2^k \alpha)$ as essentially an average of a bunch of short sums $\frac{1}{n} \sum_{k=0}^{n-1} e(2^k \beta)$). I suspect in fact that $L$ is precisely the intersection of these convex hulls, using the same sort of handwavy arguments you already have in your post. | |
| Feb 19, 2016 at 15:02 | comment | added | user9072 | I added a toplevel tag, that is one with a two-letter prefix corresponding to arXiv categories. It is recommended to use at least on such tag (in addition to more specific tags). In case I did not pick the best one, you could change it easily via an edit. | |
| Feb 19, 2016 at 15:00 | history | edited | user9072 |
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| Feb 19, 2016 at 14:44 | comment | added | Benoît Kloeckner | I think the answer is known, but don't have the time to dig up the details. Here are some pointers. There are many more invariant measures than the "Bernoullian" ones you describe. You can construct some by taking any $k$-step Markov chain on $\{0,1\}^\mathbb{N}$ and taking the distribution of a realization starting with the invariant distribution. The "thermodynamical formalism" provides a way to construct ergodic "Gibbs" invariant measures. Then, the multifractal analysis of dynamical systems typically aims at finding the Hausdorff dimension of the level sets of fonction such as $f_n$. | |
| Feb 19, 2016 at 11:32 | history | asked | Sean Eberhard | CC BY-SA 3.0 |