If $a(n+1) \ge a(n)/2$ for all $n$, then $L(A) = [-s,s]$ where $s = \sum_n a_n$.
On the other hand, if $a(n+1) < a(n)/2$ for all $n$, then $L(A)$ is a nowhere dense Cantor-type set, and you can recover $A$ from $L(A)$.
Robert Israel
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