I will begin by discussing a range of energy estimates, which are closely related to the sum-product problem over finite fields. I will then explore a number of applications including improved bounds on the few sums, many products problem, and expansion of orbits of dynamical systems generated by quadratic maps.

 

In particular, I will give a brief overview of the methods of Cilleruelo, Garaev, Ostafe, and Shparlinski (2012) and Chang (2014), in the study of polynomial orbits, highlighting the superiority of more recent techniques based on incidence geometry.

 

The talk will include results from joint works with Bryce Kerr, Thang Pham, Sophie Stevens, and Yiting Wang.