Large sets containing no copies of a given infinite sequence

Suppose $a_n$ is a real, nonnegative sequence that does not increase exponentially. For any $p<1$ we contruct a Lebesgue measurable set $E \subseteq \mathbb{R}$ which has measure at least $p$ in any unit interval and which contains no affine copy $\{x+ta_n:\ n\in\mathbb{N}\}$ of the given sequence (for any $x \in \mathbb{R}, t > 0$). We generalize this to higher dimensions and also for some ``non-linear'' copies of the sequence. Our method is probabilistic.