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: (1) There is a sequence $x_n\in H$ with $\mathrm{\parallel }{x}_n\mathrm{\parallel }=a_n$, having 0 as a weak cluster point; (2) ${\sum }_1^{\mathrm{\infty }}{a}_n^{-2}=\mathrm{\infty }$. 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.