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

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.