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Summary: Characteristic factors and pleasant extensions for some
nonconventional ergodic averages
Since Furstenberg's seminal 1977 work on the ergodic-theoretic reformulation and
proof of Szemer´edi's Theorem in additive combinatorics through an analysis of
multiple recurrence, the use of ergodic-theoretic techniques to study questions of
arithmetic combinatorics has become a rich subdiscipline in its own right. Central
in this area are certain `nonconventional' ergodic averages, such as the averages
1
N
N
n=1
Tn
1 f1 · Tn
2 f2 · · · · · Tn
k fk
corresponding to a tuple of commuting probability-preserving transformations
T1, T2, . . . , Tk on a probability space (X, , µ) and functions f1, f2, . . . , fk
L
(µ).
We will describe some recent progress on understanding the structure of these
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