Weak cluster points of a sequence and coverings by cylinders

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Let H be a Hilbert space. Using Ball's solution of the "complex plank problem" we prove that the following properties of a sequence aₙ>0 are equivalent: (1) There is a sequence xₙ ∈ H with xₙ=aₙ, having 0 as a weak cluster point; (2) ∑₁∞ aₙ⁻²=∞. 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.