A Note on the Sum-Product Problem and the Convex Sumset Problem

We provide a new exponent for the Sum-Product conjecture on $\mathbb{R}$. Namely for $A \subset \mathbb{R}$ finite, $$\max \left\{ \left\lvert A+A \right\rvert , \left\lvert AA \right\rvert \right\} \gg_{\epsilon} \left\lvert A \right\rvert ^{\frac{4}{3} + \frac{10}{4407} - \epsilon} .$$ We also provide new exponents for $A \subset \mathbb{R}$ finite and convex, namely $$\left\lvert A+A \right\rvert \gg_{\epsilon} \left\lvert A \right\rvert ^{\frac{46}{29} - \epsilon},$$ and $$\left\lvert A-A \right\rvert \gg_{\epsilon} \left\lvert A \right\rvert ^{\frac{8}{5} + \frac{1}{3440} -\epsilon} .$$