Ottawa, June 7 - 11, 2021
For example, in recent joint work with Olof Sisask, building upon ideas of Bateman and Katz, we proved a particularly strong structural result about certain kinds of large spectrum, which allowed us to obtain new bounds for sets without three-term arithmetic progressions.
In this talk I will give a survey of our current understanding of such sets, what they can look like, and will highlight some of the gaps in our knowledge, in particular some conjectures that, if solved, should yield further progress on the bounds for sets without three-term arithmetic progressions.
The crucial point here is the condition for the vectors $x_1,\dots,x_k$ in the solution $(x_1,\dots,x_k)\in A^k$ to be distinct. If we relax this condition and only demand that $x_1,\dots,x_k$ are not all equal, then the statement would follow easily from Tao's slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments.
This improves the exponent of $11/9$ by Rudnev, Shakan and Shkredov from 2018.