SCHWEITZER 1998 Problem 4

For any measurable set $H \subset R$ , we define the sequence $a_n(H)$ by the formula: $$a_n(H) = \lambda \bigg([0,1] \setminus \bigcup_{k = n}^{2n} (H + \log_2 k) \bigg)$$ where $\lambda$ denotes the Lebesgue measure and $\log_2$ denotes the binary logarithm. Prove that there is a measurable, 1-periodic, positive measure set $H \subset R$ , such that the sequence $a_n( H )$ does not belong to any space $l_p$ ( $1 \leq p < \infty$ ).