The polynomial method and restricted sums of congruence classes

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. © 1996 Academic Press, Inc.