The proposed research lies in the area of ergodic theory and has several potential implications in combinatorics. In ergodic theory, the investigator, building on his previous work, plans to carry out an in depth analysis of the limiting behavior of multiple ergodic averages along various integer sequences, for example Hardy sequences, random sequence, and sequences related to the prime numbers. The tools to be used include recent advances in the theory of characteristic factors and equidistribution results on nilmanifolds. The combinatorial implications are related to exhibiting patterns that can be found within every set of integers with positive density, thus obtaining several far reaching extensions of the celebrated theorem of Szemeredi on arithmetic progressions. Such results serve as a first step in exhibiting the same patterns within the set of prime numbers.
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