We prove that if for a finite set $A$ of integers we have $|A+A|\le K\left|A\right|$, then $A$ is contained in a generalized arithmetic progression of dimension at most $K^{1+C(\log K{)}^{-1/2}}$ and of size at most $\exp (K^{1+C(\log K{)}^{-1/2}})\left|A\right|$ for some absolute constant $C$. We also discuss a number of applications of this result.

On donne des résultats de non-densité pour les points entiers sur des variétés affines, dans l'esprit de la conjecture de Lang-Vojta. En particulier, soit $X$ une variété projective de dimension $d\ge 2$ sur un corps de nombres $K$ (resp., sur $\mathbb{C}$). Soit $H$ la somme de $2d$ diviseurs amples sur $X$ qui se coupent proprement. On montre que tout ensemble de points quasi-entiers (resp., toute courbe entière) sur $X-H$ est non Zariski-dense.

We give nondensity results for integral points on affine varieties, in the spirit of the Lang-Vojta conjecture. In particular, let $X$ be a projective variety of dimension $d\ge 2$ over a number field $K$ (resp., over $\mathbb{C}$). Let $H$ be the sum of $2d$ properly intersecting ample divisors on $X$. We show that any set of quasi-integral points (resp., any integral curve) on $X-H$ is not Zariski-dense.

Giroux showed that every contact structure on a closed $3$-dimensional manifold is supported by an open book decomposition. We extend this result by showing that the open book decomposition can be chosen in such a way that the pages are solutions to a homological perturbed holomorphic curve equation.

On a fixed smooth compact Riemann surface with boundary $(M_0,g)$, we show that, for the Schrödinger operator ${\Delta }_g+V$ with potential $V\in C^{1,\alpha }(M_0)$ for some $\alpha >0$, the Dirichlet-to-Neumann map $N{|}_{\Gamma }$ measured on an open set $\Gamma \subset \partial M_0$ determines uniquely the potential $V$. We also discuss briefly the corresponding consequences for potential scattering at zero frequency on Riemann surfaces with either asymptotically Euclidean or asymptotically hyperbolic ends.

We prove a quantitative version of the Oppenheim conjecture for inhomogeneous quadratic forms. We also give an application to eigenvalue spacing on flat $2$-tori with Aharonov-Bohm flux.