By Karl E. Petersen, Ibrahim Salama
This quantity comprises articles that describe the connections among ergodic thought and convergence, tension thought, and the idea of joinings. those papers current the historical past of every quarter of interplay, the main remarkable contemporary effects, and the at present promising strains of study. within the mixture, they're going to supply an ideal advent for an individual starting study in a single of those parts.
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Additional info for Ergodic Theory and Harmonic Analysis: Proceedings of the 1993 Alexandria Conference
Example text
Let pn denote the n-th prime number. Show that with probability 1 the random sequence A", defined by pn E A" if and only if Yn(w) = 1, is good for periodic systems. 12. ) 3. Sequences that are bad for periodic systems In this section we briefly examine sequences A = (an) that are not good for periodic systems. We may distinguish several degrees of bad behavior. It is possible that A is not good mod q for only some values of q. But maybe A is not good mod q for every q. If we know that Mt(A, f)(x) does not converge, it may mean only that we do not have convergence only for some x E Xq, but maybe there is no convergence for any x E Xq.
Let /3o E [0,1) be a point where the supremum of If I is taken, and let 0 E St be closest to i30i so we have 1/30 -)31 < 1/20t. 11). Let us introduce the random variables Zn(w) _ (Yn - v)e(n,3). We need to prove 4 I E Zn dP(w) < Ct2. 13) n A(b/q) = 1 > e(n2b/q) = q nm