Selfridge and Straus [SeSt58] also showed that the conjecture is true when 1) $k = 3$ and $|A| > 6$ or 2) $k = 4$ and $|A| > 12$. More generally, they proved that $A$ is determined by $A_k$ (and $|A|$) if $|A|$ is divisible by a prime greater than $k$.