Discretised sum-product theorems by Shannon-type inequalities

Abstract

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 $|A|={\delta }^{-\sigma },$ then $A+A$ or $AA$ 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.

arXiv comment14 pages, re-written and heavily condensed following referee suggestions. Results unchanged. Previous standalone chapter 3 removed as not necessary for main argument