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