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If $A+A+A$ contains the extremes, does it contain the middle?

Let $b \ge 1$ and $A\subseteq [0,b]$ be a set of integers (all intervals will be of integers). Write $hA := \underbrace{A + \ldots + A}_{h\text{ summands}} = \{ \sum_{i=1}^h a_i ~|~a_i \in A,\, \foral …
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