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

Abstract: 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_ε \left\lvert A \right\rvert ^{\frac{4}{3} + \frac{10}{4407} - ε} .] We also provide new exponents for $A \subset \mathbb{R} $ finite and convex, namely [ \left\lvert A+A \right\rvert \gg_ε \left\lvert A \right\rvert ^{\frac{46}{29} - ε}, ] and [ \left\lvert A-A \right\rvert \gg_ε \left\lvert A \right\rvert ^{\frac{8}{5} + \frac{1}{3440} -ε} .]