Characterizations of a Banach Space through the Strong Lacunary and the Lacunary Statistical Summabilities

In this manuscript we characterize the completeness of a normed space through the strong lacunary (N-theta) and lacunary statistical convergence (S-theta) of series. A new characterization of weakly unconditionally Cauchy series through N-theta and S-theta is obtained. We also relate the summability spaces associated with these summabilities with the strong p-Cesaro convergence summability space

On minimal additive complements of integers

Let $C,W\subseteq \mathbb{Z}$. If $C+W=\mathbb{Z}$, then the set $C$ is called an additive complement to $W$ in $\mathbb{Z}$. If no proper subset of $C$ is an additive complement to $W$, then $C$ is called a minimal additive complement. Let $X\subseteq \mathbb{N}$. If there exists a positive integer $T$ such that $x+T\in X$ for all sufficiently large integers $x\in X$, then we call $X$ eventually periodic. In this paper, we study the existence of a minimal complement to $W$ when $W$ is eventually periodic or not. This partially answers a problem of Nathanson.Comment: 13 page

Exact additive complements

Let A, B be sets of positive integers such that A + B contains all but finitely many positive integers. Sárközy and Szemerédi proved that if A(x)B(x)/x → 1, then A(x)B(x) - x → ∞. Chen and Fang considerably improved Sárközy and Szemerédi's bound. We further improve their estimate and show by an example that our result is nearly best possible. © 2016. Published by Oxford University Press. All rights reserved