In this paper, we prove that the bound \[ \max \{ |8A-7A|,|5f(A)-4f(A)| \} \gg |A|^{ \frac{ 3}{ 2} + \frac{ 1}{ 54}-o(1)} \] holds for all $A \subset \mathbb R$, and for all convex functions $f$ which satisfy an additional technical condition. This technical condition is satisfied by the logarithmic function, and this fact can be used to deduce a sum-product estimate \[ \max \{ |16A| , |A^{ (16)}| \} \gg |A|^{ \frac{ 3}{ 2} + c}, \] for some $c>0$. Previously, no sum-product estimate over $\mathbb R$ with exponent strictly greater than $3/2$ was known for any number of variables. Moreover, the technical condition on $f$ seems to be satisfied for most interesting cases, and we give some further applications. In particular, we show that \[ |AA| \leq K|A| \implies \,\forall d \in \mathbb R \setminus \{ 0 \}, \,\, |\{ (a,b) \in A \times A : a-b=d \}| \ll K^C |A|^{ \frac{ 2}{ 3}-c'}, \] where $c,C>0$ are absolute constants.
Categories: math.CO math.NT