The US claims control over all of it! After World War II the US suddenly realised sovereignity over Antarctica was divided amongst several weaker European and South American countries - as well as the USSR! The US had no say at all over possible mineral or strategic advantages in the far south.
How to resolve the issue then without upsetting the USSR or those nations the US claimed they were friendly towards? They plonked a military base smack on the South Pole itself! The Scott-Amundsen Base therefore encompasses every segment of the Antarctic. The US therefore now claims an interest in the entire continent!
Gee... here's me thinking the Antarctic Treaty System placed into effect in 1961 and encompassing "46 treaty member nations: 28 consultative and 18 acceding. Consultative (voting) members include the seven nations that claim portions of Antarctica as national territory as well as the 21 non-claimant nations that either do not recognize the claims of others, or have not stated their positions" (Source: Wiki.) was working well... but I guess not...