On the size of $k$-fold sum and product sets of integers

We prove the following theorem: for all positive integers $b$ there exists a positive integer $k$, such that for every finite set $A$ of integers with cardinality $\mathrm{\mid }A\mathrm{\mid }>1$, we have either $$\mathrm{\mid }A+\mathrm{.}\mathrm{.}\mathrm{.}+A\mathrm{\mid }\ge \mathrm{\mid }A{\mathrm{\mid }}^b$$ or $$\mathrm{\mid }A\cdot \mathrm{.}\mathrm{.}\mathrm{.}\cdot A\mathrm{\mid }\ge \mathrm{\mid }A{\mathrm{\mid }}^b$$ where $A+\mathrm{.}\mathrm{.}\mathrm{.}+A$ and $A\cdot \mathrm{.}\mathrm{.}\mathrm{.}\cdot A$ are the collections of $k$-fold sums and products of elements of $A$ respectively. This is progress towards a conjecture of Erd"os and Szemer'edi on sum and product sets.