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At primary school the first author was taught to estimate the area of a (convex) body by drawing it on a piece of graph paper, and then counting the number of (unit) squares inside. There is obviously a little ambiguity in deciding how to count the squares which straddle the boundary. Whatever the protocol, if the boundary is more-or-less smooth then the number of squares in question is proportional to the perimeter of the body, which will be small compared to the area (if the body is big enough). At secondary school the first author learnt that there are other methods to determine areas, sometimes more precise. As an undergraduate he learned that counting lattice points is often a difficult question (and that counting unit squares is “equivalent” to counting the lattice points in their bottom left-hand corner).

We give a relatively easy proof of the Erdős-Kac theorem via computing moments. We show how this proof extends naturally in a sieve theory context, and how it leads to several related results in the literature.

In these notes we discuss various sequences of numbers which are motivated by cryptographic considerations. This suggests the study of their uniform distribution and, in turn, the bounding of relevant exponential sums. Several of the bounds we give have since been quantitatively sharpened, by Garaev (Garaev, 2005) and, spectacularly so, in recent work of Bourgain (Bourgain, 2004; Bourgain, 2005).

What follows is an expanded version of my lectures at the NATO School on Equidistribution. I have tried to keep the informal style of the lectures. In particular, I have sometimes oversimplified matters in order to convey the spirit of an argument.

One of the themes of the summer school is the distribution of “special points” on varieties. In Heath-Brown’s lectures we study rational points on projective hyper-surfaces; in Ullmo’s course we study Galois orbits and Duke’s lectures deal with CM-points on the modular curve. This lecture concerns one of the earliest examples, namely torsion points on group varieties.

How are the roots of a polynomial distributed (in ℂ)? The question is too vague for if one chooses one’s favourite complex numbers z_1, z_2, ⋯, z_d then the polynomial Π^d _j=1(x - z_j) has its roots at these points.

These notes were prepared for the 2005 Summer School “Equidistribution in number theory” organized by Andrew Granville and Zeev Rudnick in Montréal. It’s a pleasure to thank them for the opportunity of giving these lectures. The aim of this text is to describe the conjectures of Manin–Mumford, Bogomolov and André–Oort from the point of view of equidistribution. This includes a discussion of equidistribution of points with small heights of CM points and of Hecke points.We tried also to explain some questions of equidistribution of positive dimensional “special” subvarieties of a given variety.

The most important analytic method for handling equidistribution questions about rational points on algebraic varieties is undoubtedly the Hardy– Littlewood circle method. There are a number of good texts available on the circle method, but the reader may particularly wish to study the books (Davenport, 2005) and (Vaughan, 1997).

We discuss Manin’s conjecture (with Peyre’s refinement) concerning the distribution of rational points of bounded height on Del Pezzo surfaces, by highlighting the use of universal torsors in such counting problems. To illustrate the method, we provide a proof of Manin’s conjecture for the unique split singular quartic Del Pezzo surface with a singularity of type D _4.