Title:
Set Images and Convexity Properties of Convolutions for Sum Sets and Difference Sets
Set Images and Convexity Properties of Convolutions for Sum Sets and Difference Sets
Author(s)
Lee, Chi-Nuo
Advisor(s)
Croot, Ernest
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Abstract
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 will focus on the properties of convolutions.
Given an abelian group $G$ and sets $A \subseteq G,$ we study the properties of the convolution for sum sets and difference sets, $1_A*1_A$ and $1_A*1_{-A}.$ Given $\bm{x} \in G^n,$ we consider its corresponding \emph{set image} of the sum set, the image of $f(A):= 1_A*1_A(\bm{x}),$ and the similarly defined set image of the difference set. We break down the study of set images into two cases, when $\bm{x}$ is independent, and when $\bm{x}$ 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 $1_A*1_A*1_A*1_A.$ We then construct different examples to better understand possible bounds for the smoothness property in the case of 2-fold convolutions $1_A*1_A.$
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Date Issued
2023-07-31
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Resource Type
Text
Resource Subtype
Dissertation