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 = 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

A Pollard Type Result for Restricted Sums

“…The next theorem is proved in [1] and states a necessary condition for the existence of nice bases with prescribed indices when the constraint of complete controllability is skipped.…”

“…If |X| =k we say that X is a k-set. In [1] we prove the following theorem: This result is an extension to an arbitrary field of a theorem proved by Pollard [4,5], for Z p =ZÂp Z, where p is a prime number. Notice that the case where t=1 is the well known Cauchy Davenport Theorem.…”

Blocks of consecutive integers in sumsets (A + B)t

“…Caldeira and Dias da Silva gave in [2] an extension of the Pollard's theorem to an arbitrary field, and in [1] an analogue for restricted sums.…”