What is the result of a 4-4 split in a US Supreme Court case?
In the case of a 4-4 tie, the decision from the lower court is left as-is, and the opinion sets no precedent that would bind other circuits.
The opinions in tied cases are generally very short, simply stating that the court was equally divided. For example, in Costco Wholesale Corp. v. Omega, S. A. 562 U.S. ___ (2010), Justice Kagan recused herself due to prior involvement with the case when she was with the Justice Department. The court split 4-4. This was the entirety of the per curiam opinion:
The judgment is affirmed by an equally divided Court.
Justice Kagan took no part in the consideration or decision of this case.
In another example, Reagan v. Abourezk, 484 U.S. 1 (1987), the court tied 3-3 (quorum is six). There was a vacancy due to the retirement of Justice Powell; Justices Blackmun and Scalia recused themselves.
History and congressional support
Edward A. Hartnett, in Ties in the Supreme Court of the United States, gives a historical account of this practice. This is a practice established by the court itself, not by direction of Congress, but Congress has "presupposed the existence of the rule of affirmance by an equally divided court". Hartnett points to 28 U.S. Code § 2109. That section relates to quorum, but notes that if the court fails to achieve quorum:
[...] the court shall enter its order affirming the judgment of the court from which the case was brought for review with the same effect as upon affirmance by an equally divided court.
In cases of original jurisdication
The situation is not so clear for Original Jurisdiction Deadlocks. In these cases, there is no lower court ruling to affirm. There have been only two original jurisdiction deadlocks. Michael Coenen summarizes:
First, in the nineteenth century case of Virginia v. West Virginia, Chief Justice Chase announced that the Justices were “equally divided on the demurrer, and equally divided also upon the order which should be made in consequence of that division.” As a result, the matter stood unresolved for nearly three years.
Second, in the twentieth-century disbarment action of In re Isserman, the Court split evenly on the question of disbarment but ordered disbarment anyway. One year later, the Court changed its mind and overruled its prior decision.
The court has a few internal things they can do to avoid ties. They could delay issuing an opinion on a case and ask for re-argument by the parties in front of the eventual nine-justice court. They could revisit the case in conference and get votes to shift around based on a more narrow or procedural ruling. In both of these situations, we wouldn't necessarily know that there was an internal deadlock.