  We study a type I see-saw scenario where the right-handed (RH) neutrinos,
responsible for the light neutrino mass generation, lie at the electroweak
scale. Under certain conditions, the strength of the charged and neutral
current weak interactions of the Standard Model particles with the heavy RH
neutrinos can be large enough to allow their production at the LHC, opening
also the possibility of observing other low energy signatures of the new
physics in the electroweak precision observables as well as in searches for
rare leptonic decays or neutrinoless double beta decay. We argue that in this
scenario the flavour structure of the neutrino Yukawa couplings is essentially
determined by the low energy neutrino parameters, leading to fairly strong
correlations among the new phenomena. In particular, we show that the present
bound on the $\mu \to e +\gamma$ decay rate makes very difficult the
observation of the heavy RH neutrinos at the LHC or the observation of
deviations from the Standard Model predictions in the electroweak precision
data. We also argue that all present experimental constraints on this scenario
still allow i) for an enhancement of the rate of neutrinoless double beta
decay, which thus can be in the range of sensitivity of the GERDA experiment
even when the light Majorana neutrinos possess a normal hierarchical mass
spectrum, and ii) for the predicted $\mu \to e+ \gamma$ decay rate to be within
the sensitivity range of the MEG experiment.
