Planar sets containing no three collinear points and non-averaging sets of integers
Let \(A\) be a set on the plane with no three collinear points, \(| A| =n\). It is proved that \[ | A+A| \gg n ( \log n)^\delta \] with any \(\delta <1/8\), but \(| A+ A| \ll n \exp c \sqrt { \log n}\) is possible. The lower bound follows from combining a result of the reviewer [Period. Math. Hung. 25, 105--111 (1992; Zbl 0761.11005)] with \textit{J. Bourgain}'s bound for sets without a 3-term arithmetic progression [Geom. Funct. Anal. 9, 968--984 (1999; Zbl 0959.11004)]. The upper estimate is based on \textit{A. P. Bosznay}'s version of Behrend's construction [Acta Math. Hung. 53, 155--157 (1989; Zbl 0682.10049)]. The lower estimate uses only that the terms of an arithmetic progression are collinear, and the author discusses the possibility of an improvement via estimating the quantity \(s_t(n)\), defined as the maximal cardinality of a subset of \(\{1, \dots , n\}\) without a nontrivial solution to each equation \(ax+by=(a+b)z\) with integers \(1\leq a,b\leq t\). In order to be useful in this context such an estimate ought to satisfy \(n/s_t(n) \ll (n/r_3(n))^t\), which the reviewer thinks to be unlikely.
scientific article; zbMATH DE number 1838933