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

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 aset membership, variantA and bset membership, variantB. For cset membership, variantA+B, νc(A,B) is the cardinality of the set of pairs (a,b) such that a+b=c. We call νc(A,B) the multiplicity of c (in A+B). Let i be a positive integer. We denote by μi(A,B) or briefly by μi the cardinality of the set of the elements of A+B that have multiplicity greater than or equal to i. Let image be a field. Let p be the characteristic of image in case of finite characteristic and ∞ if image has characteristic 0. Let A and B be finite non-empty subsets of image. We will prove that for every ℓ=1,…,min{A,B} one hasimageμ1+cdots, three dots, centered+μℓgreater-or-equal, slantedℓmin{p,A+B−ℓ}.This statement on the multiplicities of the elements of A+B generalizes Cauchy–Davenport Theorem. In fact Cauchy–Davenport is exactly inequality (a) for ℓ=1. When image inequality (a) was proved in J.M. Pollard (J. London Math. Soc. 8 (1974) 460–462); see also M.B. Nathanson (Additive number theory: Inverse problems and the geometry of sumsets, Springer, New York, 1996).