$ABC$ sum-product theorems for Katz-Tao sets

TL;DR

This paper establishes new $ABC$ sum-product theorems for Katz-Tao separated sets, allowing for different set sizes and non-concentration exponents, advancing the understanding of sum-product phenomena in fractal sets.

Contribution

The paper introduces two sharp variants of the $ABC$ sum-product theorem that relax previous size-matching constraints for Katz-Tao sets, broadening applicability.

Findings

New $ABC$ sum-product theorems for Katz-Tao sets

Results are sharp under given hypotheses

Generalizes previous sum-product results

Abstract

I prove two variants of the $ABC$ sum-product theorem for $\delta$-separated sets $A,B,C \subset [0,1]$ satisfying Katz-Tao spacing conditions. The main novelty is that the cardinality of the sets $B,C$ need not match their non-concentration exponent. The new $ABC$ theorems are sharp under their respective hypotheses, and imply the previous one.