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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
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
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.


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics