Geodesic
Sumsets of semiconvex sets
Canadian mathematical bulletin, Tome 65 (2022) no. 1, pp. 84-94

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DOI

We investigate additive properties of sets $A,$ where $A=\{a_1,a_2,\ldots ,a_k\}$ is a monotone increasing set of real numbers, and the differences of consecutive elements are all distinct. It is known that $|A+B|\geq c|A||B|^{1/2}$ for any finite set of numbers $B.$ The bound is tight up to the constant multiplier. We give a new proof to this result using bounds on crossing numbers of geometric graphs. We construct examples showing the limits of possible improvements. In particular, we show that there are arbitrarily large sets with different consecutive differences and sub-quadratic sumset size.
DOI : 10.4153/S0008439521000096
Mots-clés : sumset inequalities, convex sets of numbers, additive combinatorics 
Ruzsa, Imre; Solymosi, Jozsef. Sumsets of semiconvex sets. Canadian mathematical bulletin, Tome 65 (2022) no. 1, pp. 84-94. doi: 10.4153/S0008439521000096
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