A better than $3/2$ exponent for iterated sums and products over $\mathbb R$

Abstract: 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.