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 $(δ,σ)$-set, with $|A| = δ^{-σ},$ then $A+A$ or $AA$ has $δ$-covering number at least $δ^{-c}|A|$ for any $0 < c < \min\{σ/6, (1-σ)/6\}$ provided that $δ> 0$ is small enough.