**Title:** Configurations in sumsets

**Speaker:** John Griesmer

**Speaker Info:** University of Denver

**Brief Description:**

**Special Note**:

**Abstract:**

E. Szemeredi famously proved that every set of integers having positive upper Banach density contains arbitrarily long arithmetic progressions; Furstenberg subsequently proved the same result using ergodic theory. Refinements to the ergodic theoretic approach now provide very detailed information about the structure of sets of positive upper Banach density. Striking results due to Bergelson, Host, Kra, and Ruzsa show that sets having positive upper Banach density necessarily contain the expected density of 3-term and 4-term arithmetic progressions, but such sets may be 'deficient' in 5-term arithmetic progressions.We study sets of the form A+B := { a+b | a in A, b in B }, where A is an infinite set of integers and B has positive upper Banach density. Combining the techniques of Bergelson, Host, and Kra with spectral information about nilsystems, we obtain detailed information about the finite configurations appearing in such sets. In particular, we find an abundance of k-term arithmetic progressions for every finite k.

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