Georgia Institute of Technology College of Sciences

Many recent breakthroughs in additive combinatorics, such as results relating to Roth’s theorem or inverse sum set theorems, utilize a combination of Fourier analytical and physical methods. Physical methods refer to results relating to the physical space, such as almost-periodicity results regarding convolutions. This thesis focuses on the properties of convolutions.

Given a group G and sets A ⊆ G, we study the properties of the convolution for sum sets and difference sets, 1A ∗1A and 1A ∗1−A. Given x ∈ Gn, we study the set image of its sum set and difference set. We break down the study of set images into two cases, when x is independent, and when x is an arithmetic progression. In both cases, we provide some convexity result for the set image of both the sum set and difference set. For the case of the arithmetic progression, we prove convexity by first showing a recurrence relation for the distribution of the convolution.

Finally, we prove a smoothness property regarding 4-fold convolutions 1A ∗1A ∗1A ∗1A. We then construct different examples to better understand possible bounds for the smoothness property in the case of 2-fold convolutions 1A ∗ 1A.

Committee

Prof. Ernie Croot, Advisor

Prof. Michael Lacey

Prof. Josephine Yu

Prof. Anton Leykin

Prof. Will Perkins