Document Type
Lecture
Publication Date
10-4-2019
Abstract
This talk will consist of two seemingly unrelated topics in mathematics. A common theme in analysis, combinatorics, and number theory states that if A is a large subset of a certain group G, then the difference set A - A contains nice structures. For example, Steinhaus' theorem says that if A is a subset of positive Lebesgue measure of ℝ, then A - A contains an interval around 0. We are interested in this phenomenon in vector spaces over a finite field (��ₚ). If A is a subset of positive density of ��ₚⁿ, then A - A must contain a large subspace. Furthermore, the more sums or differences we take (e.g., A + A - A - A), the larger subspaces we are guaranteed to find. This is the content of Bogolyubov's theorem. I will talk about a bilinear analog of this theorem. The Möbius function is one of the most important functions in number theory. However, its behavior is random-like in many respects. I will talk about a function field instance of this principle, namely that the Möbius function over ��_q[t] does not correlate with linear or quadratic phases. Curiously, our main tool in establishing this is the bilinear Bogolyubov theorem above. This is joint work with Pierre-Yves Bienvenu.
Relational Format
presentation
Recommended Citation
Lê, Thái Hoàng, "Subspaces in Difference Sets and Möbius Randomness" (2019). Algebra/Number Theory Seminar. 23.
https://egrove.olemiss.edu/math_algebra_number_theory/23
