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Convex sequences may have thin additive bases. (English) Zbl 1458.11018
The authors prove that there is a $$c>0$$ such that for any $$m$$ there is a set $$A$$ of size $$n>m$$ such that the sumset $$A+A$$ contains a convex set of size $$cn^2$$. This result answers in the affirmative a question posed by P. Hegarty. Here, a set $$A\subset \mathbb{R}$$ is convex if the gaps between consecutive elements of $$A$$ are strictly increasing.