A nilpotent Roth theorem |
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Authors: | V Bergelson A Leibman |
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Affiliation: | (1) Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA (e-mails: vitaly@math.ohio-state.edu, leibman@math.ohio-state.edu), US |
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Abstract: | Let T and S be invertible measure preserving transformations of a probability measure space (X, ℬ, μ). We prove that if the group generated by T and S is nilpotent, then exists in L
2-norm for any u, v∈L
∞(X, ℬ, μ). We also show that for A∈ℬ with μ(A)>0 one has . By the way of contrast, we bring examples showing that if measure preserving transformations T, S generate a solvable group, then (i) the above limits do not have to exist; (ii) the double recurrence property fails, that
is, for some A∈ℬ, μ(A)>0, one may have μ(A∩T
-n
A∩S
-
n
A)=0 for all n∈ℕ. Finally, we show that when T and S generate a nilpotent group of class ≤c, in L
2(X) for all u, v∈L
∞(X) if and only if T×S is ergodic on X×X and the group generated by T
-1
S, T
-2
S
2,..., T
-c
S
c
acts ergodically on X.
Oblatum 19-V-2000 & 5-VII-2001?Published online: 12 October 2001 |
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Keywords: | Mathematical Subject Classification (1991): 28D15 20F18 11L15 |
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