The polynomial method and restricted sums of congruence classes Melvyn B. Nathanson Summary: The polynomial method and restricted sums of congruence classes Noga Alon Melvyn B. Nathanson Imre Z. Ruzsa § Abstract We present a simple and general algebraic technique for obtaining results in Additive Number Theory, and apply it to derive various new extensions of the Cauchy-Davenport Theorem. In particular we obtain, for subsets A0, A1, . . . , Ak of the finite field Zp, a tight lower bound on the minimum possible cardinality of {a0 + a1 + . . . + ak : ai Ai, ai = aj for 0 i < j k} as a function of the cardinalities of the sets Ai. 1 Introduction The Cauchy-Davenport Theorem, which has numerous applications in Additive Number Theory, is the following. Theorem 1.1 ([3]) If p is a prime, and A, B are two nonempty subsets of Zp, then |A + B| min{p, |A| + |B| - 1}. This theorem can be proved quickly by induction on |B|. A different proof has recently been found by the authors [1]. This proof is based on a simple algebraic technique, and its main advantage is that it J. of Number Theory 56 (1996), 404-417. Collections: Mathematics