 
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 CauchyDavenport 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 CauchyDavenport 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), 404417.
