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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 ∈ A and b ∈ B. For c ∈ A + B, ν c (A, B) is the cardinality of the set of pairs (a, b) such that a + b = c. We call ν c (A, B) the multiplicity of c (in A + B). Let i be a positive integer. We denote by µ i (A, B) or briefly by µ 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 characteristic and ∞ if F has characteristic 0. Let A and B be finite non-empty subsets of F. We will prove that for every = 1,. .. , min{|A|, |B|} one has µ 1 + • • • + µ min{p, |A| + |B| − }. (a) This statement on the multiplicities of the elements of A + B generalizes Cauchy-Davenport Theorem. In fact Cauchy-Davenport is exactly inequality (a) for = 1. When F = Z p inequality (a) was proved in

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For partition of m and A finite nonempty subset of a field we define the set of -restricted sums of m-tuples of elements of A,^m A, and using Additive Number Theory results from [J.A. Dias da Silva and Y.O. Hamidoune (1990). A note on the minimal polynomial of the Kronecker sum of two linear operators. Linear Algebra Appl., 141, 283-287; J.A. Dias da Silva and Y.O. Hamidoune (1994). Cyclic spaces for Grassmann derivatives and additive theory. Bull. London Math. Soc., 26, 140-146] we obtain a lower bound for its cardinality. Next, using results and techniques from [J.A. Dias da Silva and Y.O. Hamidoune (1990). A note on the minimal polynomial of the Kronecker sum of two linear operators. Linear Algebra Appl., 141, 283-287; J.A. Dias da Silva and Y.O. Hamidoune (1994). Cyclic spaces for Grassmann derivatives and additive theory. Bull. London Math. Soc., 26, 140-146] we obtain lower bounds for the degrees of minimal polynomials of restrictions of derivations to ranges of Young symmetrizers and to the symmetry class of tensors V , and we show that the lower bound for the cardinality of^m A can also be obtained from these lower bounds.