Sum-products, low energies and Sidon sets

A central heuristic in arithmetic combinatorics concerns an incongruence between additive and multiplicative structure in integers. This is encapsulated in a famous conjecture of Erdős–Szemerédi, which states that any finite set of integers either produces many sums or many products. While this problem remains widely open, much work has been done on this, including the breakthrough work of Bourgain–Chang on many-fold sums and products.More recently, such heuristics have been generalised from just studying the number of sums and products to analysing systems of additive and multiplicative equations, as well as considering the problem of finding large additive or multiplicative Sidon sets. In this talk, I will give an overview of the sum-product conjecture as well as its aforementioned generalisations. I will then present some of my own recent results in this direction, while also expositing some of the proof ideas contained therein.