TITLE: Estimates on invariant tori near an elliptic equilibrium point of a Hamiltonian system AUTHORS: Amadeu Delshams(1) and Pere Gutierrez(2) (1) Dept. de Matematica Aplicada I (ETSEIB), Universitat Politecnica de Catalunya, Diagonal 647, 08028 Barcelona (Spain). E-mail: amadeu@ma1.upc.es (2) Dept. de Matematica Aplicada II, Universitat Politecnica de Catalunya, Pau Gargallo 5, 08071 Barcelona (Spain). E-mail: gutierrez@ma2.upc.es ABSTRACT: We give a precise statement for KAM theorem in a neighbourhood of an elliptic equilibrium point of a Hamiltonian system. If the frequencies of the elliptic point are nonresonant up to a certain order $K\ge4$, and a nondegeneracy condition is fulfilled, we get an estimate for the measure of the complement of the KAM tori in a neighbourhood of given radius. Moreover, if the frequencies satisfy a Diophantine condition, with exponent $\tau$, we show that in a neighbourhood of radius $r$ the measure of the complement is exponentially small in $(1/r)^{1/(\tau+1)}$. We also give a related result for quasi-Diophantine frequencies, which is more useful for practical purposes. The results are obtained by putting the system in Birkhoff normal form up to an appropiate order, and the key point relies on giving accurate bounds for its terms.