Abstract
Let be a commutative group and and be nonempty finite subsets of and . Kneser’s theorem is a fundamental result in additive number theory, and it establishes that . For any subset of , write . For any , set . An important problem in additive number theory is to find a Kneser-type theorem for the restricted sumsets . In particular, more than 20 years ago V. Lev proved that if , for all (resp., ) there is at most one (resp., ) such that (resp., ), and ; then In the same paper, Lev proposed as a problem to improve to something of the form with whenever . Lev’s problem has been solved for some particular groups and some specific subsets of . However, it remains open for arbitrary groups and arbitrary large subsets of . Here, as a consequence of the main result of this paper, it is shown that if we take instead of in the lower bound of , then indeed we can take as the coefficient of something of the form with whenever .
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