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

Abstract

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 ℓ=1,…, min{|A|,|B|} one has (a) μ 1+⋯+μ ℓ⩾ℓ min{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 F= Z p 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).