Sums and products in sets of positive density

Hindman's conjecture states that for any finite coloring of the integers, there exist natural numbers x and y such that x, y, x+y, xy all have the same color. This conjecture remains open, with its difficulty stemming from the challenge of controlling arithmetic structures that simultaneously involve both addition and multiplication. In this talk, we will discuss how arithmetic configurations related to the ones appearing in Hindman’s conjecture are governed by certain local structure theorems. Our approach relies on tools and ideas from multiplicative number theory. This allows us to establish a density analogue of a special case of a theorem of Moreira and resolve a conjecture of Moreira.