The sum-product conjecture is false for real numbers

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 $\asymp \log \mid A\mid$) such that $$\max (\mid A+A\mid ,\mid AA\mid )\le \mid A{\mid }^{2-c}$$ where $c>0$ is an absolute constant. We also disprove the many sums and products conjecture by constructing, for any $k\ge 3$, arbitrarily large $A\subset \mathbb{R}$ such that $$\max (\mid kA\mid ,\mid A^{(k)}\mid )\le \mid A{\mid }^{C\frac{\log k}{\log \log k}}$$ 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.