The invariant polynomials degrees of the Kronecker sum of two linear operators and additive theory

Let G be an abelian group. Let A and B be finite non-empty subsets of G. By A + B we denote the set of all elements a + b with a is an element of A an d b is an element of B. For c is an element of A + B, v(c)(A, B) is the car dinality of the set of pairs (a, b) such that a + b = c. We call v(c)(A, B) the multiplicity of c tin A + B). Let i be a positive integer. We denote by mu(i) (A, B) or briefly by mu(i) the cardinality of the set of the elements of A + B that have multiplicity greater than or equal to i. Let F be a field. Let p be the characteristic of F in case of finite charac teristic and infinity if F has characteristic 0. Let A and B be finite non- empty subsets of F. We will prove that for every l = 1,...,min{\A], \B\} one has mu(1) +...+ mu l greater than or equal to l min{p, \A\ + \B\ - l} This statement on the multiplicities of the elements of A + B generalizes C auchy-Davenport Theorem. In fact Cauchy-Davenport is exactly inequality (a) for l = 1. When F = Z(p) inequality (a) was proved in J.M. Pollard (J. Lon don Math. Sec. 8 (1974) 460-462); see also M.B. Nathanson (Additive number theory: Inverse problems and the geometry of sumsets, Springer, New York, 1 996). (C) 2000 Elsevier Science Inc. All rights reserved.