Abstract. We say that a sequence (xn) n≥1 in [0, 1) has Poissonian pair correlations iffor all s > 0. In this note we show that if the convergence in the above expression is-in a certain sense-fast, then this implies a small discrepancy for the sequence (xn) n≥1 . As an easy consequence it follows that every sequence with Poissonian pair correlations is uniformly distributed in [0, 1).

Abstract. Let S be a bounded, Riemann measurable set in R d , and Λ be a lattice. By a theorem of Fuglede, if S tiles R d with translation set Λ, then S has an orthogonal basis of exponentials. We show that, under the more general condition that S multitiles R d with translation set Λ, S has a Riesz basis of exponentials. The proof is based on Meyer's quasicrystals.

We study the asymptotic behaviour of the sequence of sine products P n (α) = n r=1 |2 sin πrα| for real quadratic irrationals α. In particular, we study the subsequence Q n (α) = qn r=1 |2 sin πrα|, where q n is the nth best approximation denominator of α, and show that this subsequence converges to a periodic sequence whose period equals that of the continued fraction expansion of α. This verifies a conjecture recently posed by Mestel and Verschueren in [15].

Abstract. We study bounded remainder sets with respect to an irrational rotation of the d-dimensional torus. The subject goes back to Hecke, Ostrowski and Kesten who characterized the intervals with bounded remainder in dimension one.First we extend to several dimensions the Hecke-Ostrowski result by constructing a class of d-dimensional parallelepipeds of bounded remainder. Then we characterize the Riemann measurable bounded remainder sets in terms of "equidecomposability" to such a parallelepiped. By constructing invariants with respect to this equidecomposition, we derive explicit conditions for a polytope to be a bounded remainder set. In particular this yields a characterization of the convex bounded remainder polygons in two dimensions. The approach is used to obtain several other results as well.

We study sets of bounded remainder for the twodimensional continuous irrational rotation ({x 1 + t}, {x 2 + tα}) t 0 in the unit square. In particular, we show that for almost all α and every starting point (x 1 , x 2 ), every polygon S with no edge of slope α is a set of bounded remainder. Moreover, every convex set S whose boundary is twice continuously differentiable with positive curvature at every point is a bounded remainder set for almost all α and every starting point (x 1 , x 2 ). Finally we show that these assertions are, in some sense, best possible. Date