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Adding distinct congruence classes modulo a prime

Summary: Adding distinct congruence classes
modulo a prime
Noga Alon
Melvyn B. Nathanson
Imre Ruzsa
1 The Erdos-Heilbronn conjecture
The Cauchy-Davenport 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, Erdos-Graham [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 Erdos-Heilbronn conjecture that uses only the most elementary
properties of polynomials. The method, in fact, yields generalizations of both
the Erdos-Heilbronn conjecture and the Cauchy-Davenport theorem.
2 The polynomial method


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


Collections: Mathematics