A generalization of Sárközy’s theorem in function fields
Document Type
Lecture
Publication Date
4-9-2025
Abstract
Sárközy’s theorem says that if A ⊂ ℤ has positive upper asymptotic density, then there are distinct a1, a2 ∈ A and n ∈ ℤ such that a1 − a2 = n2. The same is true if n2 is replaced by F(n) for any polynomial F ∈ ℤ [x] with constant term zero. Using the Croot–Lev–Pach polynomial method, Green proved an �� q[t]-analog of Sárközy’s theorem with strong quantitative bounds, but required a technical condition on the number of roots of the polynomial F ∈ �� q[t][x]. This condition was recently removed by Li and Sauerman. We generalize Green’s argument to accommodate equations in more variables in �� q[t], while pointing out that the technical condition can be removed by means of a simple observation.
Relational Format
presentation
Recommended Citation
Wathodkar, Gauree, "A generalization of Sárközy’s theorem in function fields" (2025). Combinatorics Seminar. 121.
https://egrove.olemiss.edu/math_combinatorics/121