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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.

MSC:
11B13 Additive bases, including sumsets
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Full Text: DOI Euclid
References:
[1] 10.1007/BF01949064 · Zbl 0611.10037
[2] 10.1017/S0963548311000277 · Zbl 1306.11013
[3] 10.1134/S0081543815040185 · Zbl 1383.11014
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