The sum-product conjecture is false for real numbers

Abstract

We disprove the sum-product conjecture for real numbers by constructing arbitrarily large $A\subset \mathbb{R}$ (whose elements are algebraic integers in a number field of degree $\asymp \log\lvert A\rvert$) such that \[\max(\lvert A+A\rvert ,\lvert AA\rvert)\leq \lvert A\rvert^{2-c}\] where $c>0$ is an absolute constant. We also disprove the many sums and products conjecture by constructing, for any $k\geq 3$, arbitrarily large $A\subset \mathbb{R}$ such that \[\max(\lvert kA\rvert,\lvert A^{(k)}\rvert)\leq \lvert A\rvert^{C\frac{\log k}{\log\log k}}\] for some constant $C>0$. We obtain similar constructions for $p$-adics, finite fields, and function fields in positive characteristic, and also obtain new lower bounds for the number of solutions to linear equations in a multiplicative group and the number of solutions to the unit equation in sufficiently many variables.