Erdős–Szemerédi conjecture with exponent arbitrarily close to one

Prove that for every real number \(0<\epsilon<1\), there exists a threshold such that every sufficiently large finite set \(A\subset\mathbb{Z}\) satisfies \(|A+A|+|AA|>|A|^{1+\epsilon}\), thereby showing that either the sum-set or the product-set is nearly as large as possible.

Background

The paper introduces sum-product phenomena through the Erdős–Szemerédi theorem, which asserts that finite subsets of the integers cannot simultaneously possess strong additive and multiplicative structure. It records the best currently known exponent as $\epsilon=1/3+2/1167$, due to Rudnev and Stevens.

The unresolved conjecture asks whether the exponent can be improved to every value below one. If true, this would imply that for every sufficiently large finite integer set, at least one of its sum-set or product-set has size almost quadratic in the size of the original set. Although the paper studies an analogous discretised sum-product problem for Ahlfors-regular sets, this integer-set conjecture is stated as broader background and is not resolved by the paper.