Recently Konyagin and Shkredov improved (arxiv:1602.03473) the $4/3$ exponent in the sum-products estimate $|A+A|+|A\cdot A|\gg |A|^{4/3}, A\subset \mathbb{R}$, by $4/3+5/9813$. This appears to be much harder than one-page proof for $4/3$.
Fedor Petrov
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