Recently Konyagin and Shkredov improved the exponent of $4/3$ in the sum-products estimate in $\mathbb{R}$, namely that $|A+A|+|A\cdot A|\gg |A|^{4/3-o(1)}$ for every $A\subset \mathbb{R}$, to $4/3+5/9813$. This appears to be much harder than the short proof for $4/3-o(1)$ by Solymosi.
The conjecture of Erdős is that the exponent approaches 2.