Generalised sum-product phenomenon and a question of Bremner

The sum-product phenomenon concerns an incongruence between additive and multiplicative structure over finite sets of integers. Results of this type often involve showing that a finite set of integers either gives rise to many pairwise sums or many pairwise products. Such results have been studied in various other settings such as complex numbers, finite fields and algebraic groups.  

In joint work with Joseph Harrison and Harry Schmidt, we prove a sum-product type estimate in the general setting of 1-dimensional algebraic groups over complex numbers. This allows us to say something towards a question of Bremner concerning the longest arithmetic progression contained in the set of x-coordinates of rational points on an elliptic curve.  

In this talk, I will provide a gentle introduction to the sum-product phenomenon, mention our sum-product type results in the setting of algebraic groups as well as their connection to the above question of Bremner.