Discretised sum-product theorems by Shannon-type inequalities

TL;DR

This work delivers explicit, near-optimal quantitative bounds for the discretised sum-product (discretised ring) problem in one dimension by marrying fractal-geometry lower bounds with Shannon-type information inequalities. The authors leverage a Marstrand-type projection framework (via Orponen–Shmerkin–Wang) and submodular entropy inequalities to show that if a set behaves like a $\delta^{-\sigma}$-dimensional object, then at least one of $A+A$ or $AA$ must exhibit a dimension gain of order $c$ with $c$ close to $\min\{\sigma/6,(1-\sigma)/6\}$. They provide precise $(\delta,\sigma,C)$-set bounds and derive corollaries for the discretised ring theorem, along with explicit examples demonstrating both the necessity and sharpness of the bounds in various parameter regimes. The results have implications for related fractal-projection questions and contribute explicit constants to a historically qualitative area, offering a framework that combines geometric measure theory with information-theoretic tools. Overall, the paper advances a principled, quantitative approach to sum-product phenomena in fractal settings, with potential applications to distance sets and projection problems in fractal geometry.

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 $(δ,σ)$-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.

Theorems & Definitions (34)

• Definition 1.4: $(\delta,\sigma, C)$-set
• Theorem 1.5
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10
• Remark 1.12
• Remark 1.13
• Lemma 2.2
• proof
• Definition 2.4