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Complex Analysis and Geometry Seminar

Recent developments in discrete harmonic analysis

Mariusz Mirek, Kings College London

Location:  HILL 705
Date & time: Friday, 22 September 2017 at 10:30AM - 11:30AM

Abstract  In the first part of the talk we will be concerned with the problem of existence of infinitely many arithmetic progressions of length at least three in subsets which have vanishing density in the set of  prime numbers. Our principal example will be the set of Piatetski--Shapiro prime numbers.

In the second part we will explain connections of the problem raised above with some questions in the pointwise ergodic theory. Specifically, we will see the usefulness of r-variational estimates in pointwise convergence problems.

Finally, I would like to mention about some problem in pointwise ergodic theory which led us to study dimension-free bounds for maximal functions and r-variations corresponding to the discrete Hardy--Littlewood averaging operators defined over the cubes in \(mathbbZ^d\).

The last part is joint project with J. Bourgain, E.M. Stein and B. Wrobel.  

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