 
Summary: Adding distinct congruence classes
modulo a prime
Noga Alon
Melvyn B. Nathanson
Imre Ruzsa
1 The ErdosHeilbronn conjecture
The CauchyDavenport theorem states that if A and B are nonempty sets of
congruence classes modulo a prime p, and if A = k and B = l, then the
sumset A + B contains at least min(p, k + l  1) congruence classes. It follows
that the sumset 2A contains at least min(p, 2k  1) congruence classes. Erdos
and Heilbronn conjectured 30 years ago that there are at least min(p, 2k  3)
congruence classes that can be written as the sum of two distinct elements of
A. Erdos has frequently mentioned this problem in his lectures and papers (for
example, ErdosGraham [4, p. 95]). The conjecture was recently proven by
Dias da Silva and Hamidoune [3], using linear algebra and the representation
theory of the symmetric group. The purpose of this paper is to give a simple
proof of the ErdosHeilbronn conjecture that uses only the most elementary
properties of polynomials. The method, in fact, yields generalizations of both
the ErdosHeilbronn conjecture and the CauchyDavenport theorem.
2 The polynomial method
