Samantha Sandberg
The Ohio State University
Title
Arithmetic Progressions in Fractal Sets of Sufficient Thickness
Abstract
We consider the conditions required on a set that guarantee it contains arithmetic progressions. Szemeredi proved the existence of arithmetic progressions in subsets of the natural numbers with positive upper density. In the fractal setting, it is known by Maga and Keleti that full Hausdorff dimension is not enough to guarantee the existence of a $3$-term arithmetic progression in subsets of d-dimensional Euclidean space; however, it turns out that Fourier decay coupled with nearly full Hausdorff dimension is sufficient for the existence of arithmetic progressions, as shown by Laba and Pramanik. In this talk, we consider another notion of size: Newhouse thickness. It is known that thickness larger than $1$ is enough in the real line to guarantee the existence of a $3$-term arithmetic progression. In higher dimensions, Yavicoli showed that it takes thickness larger than $10^8$, along with some additional assumptions, to guarantee a $3$-point configuration. We give the first result in higher dimensions showing the existence of $3$-term arithmetic progressions in sets of thickness larger than $2/(1-2r)$, where r is a constant dependent on the set.
