Corrections

Section 5.5, Page 213, line -13

In problem 3, apart from assuming that no four points induce a parallelogram, we have to assume that no four points are on a circle, no three on a line. (Otherwise,

we could take a system of equidistant points along a circle. Adrian Dumitrescu has also proved that there are only n^2/logn distinct distances among the points (i,i^2), i=1,2,...,n, and this set meets all the requirements. Therefore, the modified problem should read as follows.

Is it true that for every positive epsilon, the number of distinct distances between any set of n points in the plane, no three of which are collinear, no four of which are cocircular or determine a parallelogram, satisfies v^{no-parallel}(n) > Omega(n^{2-epsilon})?

Bibliography: A. Dumitrescu: On distinct distances among points in general position and other related problems, Period. Math. Hungar. 57 (2008), no. 2, 165–176.