I will explain the content and some ideas from the paper of Hardy and Littlewood, Some Problems of Diophantine Approximation: The Lattice-Points of a Right-Angled Triangle. Proc. London Math. Soc. S2-20 no. 1, 15.

The talk will be about a homogeneous dynamics proof of classical theorems of Khintchin and Groshev, and their generalizations, proved in Logarithm laws for flows on homogeneous spaces, by Dmitry Kleinbock and Gregory Margulis, Inv. Math. 138 (1999). Suggested reading: it would be good to recall the statement of the Khintchin theorem and Borel-Cantelli lemma. A good starting point is wikipedia.

The talk will focus on the construction of cut and project sets which are not bounded displacement to lattices. Based on this paper.

This is the first of two talks, it will introduce the concept of singular vectors and their dynamical interpretation, Hausdorff dimension and how it is computed, statement of the main result. Based on this paper.

In this talk the upper bound in the theorem of Cheung and Chevallier will be proved, showing that the set of singular pairs in the plane has dimension 1.5.

The talk will introduce the Littlewood conjecture (still open), its dynamical formulation, and the work of Cassels and Swinnerton-Dyer who proved the conjecture for some cubic numbers. This is based on Cassels, J. W. S.; Swinnerton-Dyer, H. P. F. On the product of three homogeneous linear forms and the indefinite ternary quadratic forms. Philos. Trans. Roy. Soc. London. Ser. A. 248, (1955). 73

A set Y in R^d is called a dense forest if it is uniformly discrete and for every positive \epsilon there is a T=T(\epsilon) such that: for every x in R^d and a direction v the line segment of length T from x in direction v is \epsilon-close to Y. These objects arise as a natural weakening of the Danzer problem, and also has applications in quasi-conformal geometry. I plan to present two constructions of dense forests. The first is due to Chris Bishop (following an idea of Yuval Peres) - it is defined in the plane and it uses the Diophantine properties of the golden ratio. The second appears in a recent paper of Barak Weiss and myself. We use Ratner's theorem to construct very unique R^d-actions on some compact manifolds (of the form G/\Gamma), which in turns ensure that the sets of return times to a section are dense forests.

The relevant papers are: (-) Bishop A set containing rectifiable arcs QC-locally but not QC-globally.

(-) Solomon, Weiss Dense forests and Danzer sets.