Phase diagram in 2D Fr\"ohlich model of metal at arbitrary carrier density: pseudogap versus doping

It is shown that in 2D system of the fermions with simplest indirect boson-induced attraction (through the Einstein phonon exchange as an example) along with the normal and superconducting phases there arises a new (called "abnormal normal" or pseudogap) one where the absolute value of the order parameter is finite but its phase is a random quantity. It is important that this new phase really exists at low carriers density only, i.e. it shrinks when doping increases. The relevance of the results obtained with dependence of pseudogap on doping in high-temperature superconductors is speculated.