In this paper the following improvement on Freiman's theorem on set addition is obtained (see Theorems 1 and 2 in Section 1).

Let $A\subset \mathbb{Z}$ be a finite set such that . Then A is contained in a proper d-dimensional progression P, where $d\le \left[\alpha -1\right]$ and .

Earlier bounds involved exponential dependence in α in the second estimate. Our argument combines I. Ruzsa's method, which we improve in several places, as well as Y. Bilu's proof of Freiman's theorem.

Nous établissons la conjecture de Manin dans le cas particulier des surfaces V de del Pezzo déployées de degré 5 sur ℚ. Autrement dit, nous montrons que, pour un ouvert $U\subset V$, on a

$$N_{U(Q)}(B):=\text{card}\{P\in U(Q):h(P)\le B\}\sim CB{(\log B)}^4(B\to +\infty )$$.

La constante C est conforme à l'expression conjecturée par E. Peyre.

We state Manin's conjecture in the particular case of the split del Pezzo's surfaces of degree 5 over $Q$. We show that, for an open set $U\subset V$,

$$N_{U(Q)}(B):=\text{card}\{P\in U(Q):h(P)\le B\}\sim CB{(\log B)}^4(B\to +\infty )$$

The constant C is the one conjectured by E. Peyre.

We develop a theory of geometrically controlled branched covering maps between metric spaces that are generalized cohomology manifolds. Our notion extends that of maps of bounded length distortion, or BLD-maps, from Euclidean spaces. We give a construction that generalizes an extension theorem for branched covers by I. Berstein and A. Edmonds. We apply the theory and the construction to show that certain reasonable metric spaces that were shown by S. Semmes not to admit bi-Lipschitz parametrizations by a Euclidean space nevertheless admit BLD-maps into Euclidean space of same dimension.

In this paper, we prove that the depths of irreducible admissible representations are preserved by the local theta correspondence for any type I reductive dual pairs over a nonarchimedean local field.