Weak cluster points of a sequence and coverings by cylinders

Анотація

Let $H$ be a Hilbert space. Using Ball’s solution of the "complex plank problem" we prove that the following properties of a sequence $a_n>0$ are equivalent:
1. There is a sequence $x_n\in H$ with $||x_n||=a_n$, having 0 as a weak cluster point;
2.$\sum_{1}^{\infty}a_n^{-2}=\infty$.
Using this result we show that a natural idea of generalization of Ball's "complex plank" result to cylinders with $k$-dimensional base fails already for $k=3$. We discuss also generalizations of "weak cluster points" result to other Banach spaces and relations with cotype.