The invariant polynomials degrees of the Kronecker sum of two linear operators and additive theory (Q1583720)

Let \(A\), \(B\) be finite nonempty subsets of a field \(\mathbb{F}\). Write \(A+ B=\{a+b: a\in A, b\in B\}\) and denote by \(\mu_i= \mu_i(A,B)\) \((i= 1,2,\dots)\) the number of elements of \(A+B\) that can be represented in the form \(a+b\) in at least \(i\) ways. Let \(p\) (\(=\infty\) or a prime) be the additive order of the nonzero elements of \(\mathbb{F}\). The authors prove that \[ \sum^\ell_{i=1} \mu_i\geq \ell\min\{p,|A|+|B|- \ell\} (\ell= 1,\dots, \min\{|A|,|B|\}). \] This gives the Cauchy-Davenport Theorem: \(|A+B|\geq \min\{p,|A|+|B|- 1\}\), when \(\ell= 1\) and \(\mathbb{F}= \mathbb{F}_p\). The full result over \(\mathbb{F}_p\) was proved by \textit{J. M. Pollard} [J. London Math. Soc., II. Ser. 8, 460-462 (1974; Zbl 0322.10024)]. The proof, which is quite different from Pollard's, rests on the following observation: let \(V\), \(W\) be vector spaces of dimensions \(|A|\), \(|B|\) and let \(f\), \(g\) be linear transformations on \(V\), \(W\) with spectra \(A\), \(B\); then the spectrum of the Kronecker sum \(f\otimes I_W+ I_V\otimes g\) is the family \((a+b)_{(a,b)\in A\times B}\). Quite detailed calculations with invariant factors are required in the course of the proof.