Speaker: Igor Shparlinski, Macquarie University
We will give a survey of several recent results about the distribution of points (x, x^*) where x^* \in [1, m] is defined by the congruence x x^* = 1 (mod m) (for (x,m)=1). Clearly, bounds of Kloosterman sums immediately provide a series of nontrivial results about such points. However, we concentrate on some other methods which quite surprisingly come into play. In particular, we show that the behaviour of the points on the curve xy = 1 (mod m) is not the same as for a "random curve" f(x,y) = 1 (mod m).
We will also mention some multidimensional generalisations and further applications.