Weak cluster points of a sequence and coverings by cylinders

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:

• There is a sequence $x_n \in H$ with $\|x_n\|=a_n$, having 0 as a weak cluster point;
• $\sum_1^\infty a_n^{-2}=\infty$.

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.