Read Rogers v. Bellei 401 U.S. 815 (1971) and Afroyim v. Rusk, 387 U.S. 253 (1967) in tandem.
The Supreme Court overturned portions of an act of Congress which had listed various actions and had said that the performance of any of these actions could be taken as conclusive, irrebuttable proof of intent to give up U.S. citizenship. However, the Court ruled that a person's intent to give up citizenship could be established through a standard of preponderance of evidence (i.e., more likely than not) rejecting an argument that intent to relinquish citizenship could only be found on the basis of clear, convincing and unequivocal evidence.
SCOTUS also held in Bellei that it was legal and proper for Congress to establish rules of action necessary to retain one's citizenship. In this case, the requirement was for the spending of a period of time in the US after birth, which Bellei failed to do and subsequently lost his citizenship.