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The Cyclic Surgery Theorem and Moser's work on surgeries on torus knots imply that for any non-trivial knot in S³, there are at most two integer surgeries that produce a lens space. This paper investigates how many positive integer surgeries on a given knot in S³ can produce a manifold rational homology cobordant to a lens space. Tools include Greene and McCoy's work on changemaker lattices which come from Heegaard Floer d-invariants, and Aceto-Celoria-Park's work on rational cobordisms and integral homology which is based on Lisca's work on lens spaces.
Antony T. H. Fung (Sun,) studied this question.