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by M Walsh

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...by Tao [23]. For r = 1 this was shown much earlier by von Neumann [22] and for r = 2 by Conze and Lesigne [8]. On the other hand, far reaching generalizations were later given by Austin [1] and Walsh =-=[26]-=-. However, it is an interesting problem to quantify the convergence of the sequence (1.1) by controlling the number of its jumps of a certain size. Such a result in the case of a single transformation...

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Tim Austin

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...rrence phenomenon, when d = 2. The first of these conclusions can be extended to arbitrary d along the lines of Walsh’s recent proof of convergence for polynomial nilpotent non-conventional averages (=-=[Wal12]-=-). Theorem A. In the setting above, the functional averages Λn(f1, . . . , fd) converge in the norm of L2(µ) for all f1, . . . , fd ∈ L∞(µ). Zorin-Kranich has made the necessary extensions to Walsh’s ...

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