A Steinhaus type theorem

Abstract

The sequence space Λ r , r ≥ 1 {\Lambda _r},r \geq 1 being a fixed integer, is defined as \[ Λ r = { x = { x k } ∈ l ∞ , x k ∈ K , k = 0 , 1 , 2 , … , | x k + r − x k | → 0 , k → ∞ } , {\Lambda _r} = \left \{ {x = \left \{ {{x_k}} \right \} \in {l_\infty },{x_k} \in K,k = 0,1,2, \ldots ,\left | {{x_{k + r}} - {x_k}} \right | \to 0,k \to \infty } \right \}, \] where K K is a complete, nontrivially valued field and l ∞ {l_\infty } is the space of bounded sequences with entries in K K . In this paper, it is proved that given a regular matrix A = ( a n k ) , a n k ∈ K = R A = ({a_{nk}}),{a_{nk}} \in K = {\mathbf {R}} or C {\mathbf {C}} , there exists a sequence in Λ r − ∪ i = 1 r − 1 Λ i {\Lambda _r} - \cup _{i = 1}^{r - 1}{\Lambda _i} which is not A A -summable. This is an improvement of the well-known Steinhaus theorem. It is, however, shown that this result fails to hold when K K is a complete, nontrivially valued, nonarchimedean field, whereas it is known that the Steinhaus theorem continues to hold.