Discretised sum-product theorems by Shannon-type inequalities

By making use of arithmetic information inequalities, we give a strong quantitative bound for the discretised ring theorem. In particular, we show that if $A\subset [1,2]$ is a $(\delta ,\sigma )$-set, with $\mathrm{\mid }A\mathrm{\mid }={\delta }^{-\sigma },$ then $A+A$ or $AA$ has $\delta$-covering number at least ${\delta }^{-c}\mathrm{\mid }A\mathrm{\mid }$ for any $0<c<\min \{\sigma \mathrm{/}6,(1-\sigma )\mathrm{/}6\}$ provided that $\delta >0$ is small enough.