The Uniformed and Overseas Citizens Absentee Voting Act (UOCAVA) affords the right to vote in Federal elections (President, Vice President, U.S. Senator, U.S. Representative, or non-voting delegate from DC, PR, GU, etc.) to citizens of the United States who have LEFT the "United States" and moved (temporarily or permanently) to a place that is NOT in the United States.
UOCAVA defines the term "United States," when used in a geographical sense, as the 50 states, the District of Columbia, the Commonwealth of Puerto Rico, the United States Virgin Islands, Guam, and American Samoa.
So if you are a civilian and citizen of the United States, and you move from the state of New York to the Commonwealth of Puerto Rico, are you eligible to vote in New York by absentee ballot under UOCAVA? The answer is NO. You have NOT technically left the United States. For UOCAVA purposes, Puerto Rico is just as much part of the United States as New York is, and so you will vote as a resident of Puerto Rico.