Document Type
Lecture
Publication Date
10-26-2022
Abstract
We discuss two related notions of ‘approximate subgroups’ inside finite sets of integers: Bohr sets, which capture simultaneous diophantine approximation, and symmetric generalized arithmetic progressions (GAPs). For example, let x ∈ ℕ, d ∈ ℕ, and consider the following pair of questions: 1. For fixed λ₁, ..., λ_d ∈ ℝ, how small can we make max{|λ₁n|_��, ..., |λ_d n|_��} for 1 ≤ n ≤ N, where |⋅|_�� denotes distance to the nearest integer? 2. How large can a set of the form {x₁l₁ + ... + x_d l_d : -L_i ≤ l_i ≤ L_i} ⊆ [N, N] be before it is guaranteed to contain a perfect square? Our discussions range from classical facts like the Kronecker approximation theorem and Linnik's theorem to a recent breakthrough result of Maynard and its potential future applications. Along the way, we survey results including previous joint work with Lyall and Croot.
Relational Format
presentation
Recommended Citation
Rice, Alex, "Generalized Arithmetic Progressions and Diophantine Approximation by Polynomials" (2022). Algebra/Number Theory Seminar. 14.
https://egrove.olemiss.edu/math_algebra_number_theory/14
