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We prove that if for a finite set of integers we have , then is contained in a generalized arithmetic progression of dimension at most and of size at most for some absolute constant . 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 une variété projective de dimension sur un corps de nombres (resp., sur ). Soit la somme de diviseurs amples sur qui se coupent proprement. On montre que tout ensemble de points quasi-entiers (resp., toute courbe entière) sur 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 be a projective variety of dimension over a number field (resp., over ). Let be the sum of properly intersecting ample divisors on . We show that any set of quasi-integral points (resp., any integral curve) on is not Zariski-dense.
Giroux showed that every contact structure on a closed -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 , we show that, for the Schrödinger operator with potential for some , the Dirichlet-to-Neumann map measured on an open set determines uniquely the potential . We also discuss briefly the corresponding consequences for potential scattering at zero frequency on Riemann surfaces with either asymptotically Euclidean or asymptotically hyperbolic ends.