### A Roth type result for dense subsets of the integer lattice

- Series
- Additional Talks and Lectures
- Time
- Friday, October 11, 2024 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Akos Magyar – The University of Georgia

Let A be a subset of the integer lattice of positive upper density. Roth' theorem in this setting states that there are points x,x+y,x+2y in A with the length of the gap y arbitrary large. We show that the lengths of the gaps y contain an infinite arithmetic progression, as long as one measures the length in lp for p>2 even, while this not true for the Euclidean distance.

Such results have been previously obtained in the continuous settings for measurable subsets of Euclidean spaces using methods of time-frequency analysis, as opposed our approach is based on some ideas from additive combinatorics such as uniformity norms and arithmetic regularity lemmas. If time permits, we discuss some other results that can be obtained similarly.