Decoupling for Cantor sets on the parabola

Decoupling estimates aim to study the “amount of cancellation” that can occur when we add up functions whose Fourier transforms are supported in different regions of space. In this talk I will describe decoupling estimates for a Cantor set supported in the parabola. I will discuss how both curvature and sparsity (or lack of arithmetic structure) can separately give rise to decoupling estimates, and how these two sources of “cancellation” can be combined to obtain improved estimates for sets that have both sparsity and curvature. No knowledge of what a decoupling estimate is will be assumed. Based on joint work with Alan Chang, Rachel Greenfeld, Asgar Jamneshan, José Madrid and Zane Li