If $A \subseteq \mathbb{N}$ has unbounded dyadic-shell counts and $\sum_{n \in A} \|\theta n\| = \infty$ for every $0 < \theta < 1$, must $A$ be complete - is every sufficiently large integer a sum of distinct elements of $A$?

The hypothesis ¬ Summable (fun n : A ↦ distToNearestInt (θ * n)) used below says exactly that the partial sums of ‖θ n‖ over n ∈ A diverge, which is the form the linked proof uses. distToNearestInt is nonnegative, so this is an instance of not_summable_subtype_iff_tendsto_sum_indicator.