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_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} .]

💡 Research Summary

The paper addresses two long‑standing conjectures in additive combinatorics: the sum‑product conjecture of Erdős and Szemerédi for real sets, and the convex sumset conjecture of Erdős for convex subsets of the reals. The author provides modest but non‑trivial improvements over the best known exponents.
For an arbitrary finite set (A\subset\mathbb{R}) the new bound is
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