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Pointwise Ergodic Theorems for Nonconventional L^1 Averages


A topic of classical interest in ergodic theory is extending Birkhoff's Pointwise Ergodic Theorem to various classes of nonconventional ergodic averages. Previous methods had only established the pointwise behavior of many such averages when applied to functions in L^p with p>1, leaving open the important case of L^1.

In this thesis, I adapt and extend the techniques of Urban and Zienkiewicz and Buczolich and Mauldin, whose recent L^1 results have renewed interest in the subject, in order to prove several new results on the pointwise convergence of nonconventional averages of L^1 functions.

The first result (Theorem 1.3) proves that averages along certain Bernoulli random sequences (generated by independent {0,1}-valued random variables) satisfy a pointwise ergodic theorem in L^1, with probability 1. The second (Theorem 1.4) proves that averages along n^d for d≥2, or along the sequence of primes, do not satisfy a pointwise ergodic theorem in L^1. (The result for n^2 was first shown by Buczolich and Mauldin; the others are new.) The third (Corollary 1.9) shows that, given a Fourier decay condition on a sequence of probability measures on the integers, some subsequence of the corresponding nonconventional averages will satisfy a pointwise ergodic theorem in L^1.

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