New results on sums and products in $\mathbb R$

We improve previous sum-product estimates in $\mathbb R$; namely, we prove the inequality $\max\{|A+A|,|AA|\}\gg|A|^{4/3+c}$, where $c$ is any number less than $5/9813$. New lower bounds for sums of sets with small product set are found. We also obtain results on the additive and multiplicative energies; in particular, we improve a result of Balog and Wooley.