Samuel Mansfield

Additive growth between linearly independent analytic functions

The Erdos--Szemeredi sum-product problem conjecture states that additive and multiplicative structure shouldn't be able to coexist within a finite subset of real numbers A. Specifically, it asks for results in the form of the inequality max{|A+A|,|AA|} >> |A|^{1+c}, where 0 < c < 1 is some positive constant. It was first proved in this form by Hegyvari, with the best known value for c being given by Rudnev--Stevens. Since the product set AA is in bijection with the sumset log(A)+log(A), it makes sense to study, more generally, the above problem with AA replaced by f(A)+f(A) for any convex/concave funcion f. This has motivated a flurry of research into similar problems involving convex functions, which has found interesting applications in both arithmetic combinatorics and discrete geometry. In this talk, we will extend some of these ideas (particularly those of Bradshaw and Hanson--Roche-Newton--Rudnev) to families of analytic functions with linearly independent derivatives.