  A multi-purpose fixed-target experiment using the proton and lead-ion beams
of the LHC was recently proposed by Brodsky, Fleuret, Hadjidakis and Lansberg,
and here we concentrate our study on some issues related to the spin physics
part of this project (referred to as AFTER). We study the nucleon spin
structure through $pp$ and $pd$ processes with a fixed-target experiment using
the LHC proton beams, for the kinematical region with 7 TeV proton beams at the
energy in center-of-mass frame of two nucleons $\sqrt{s}=115$ GeV. We calculate
and estimate the $\cos2\phi$ azimuthal asymmetries of unpolarized $pp$ and $pd$
dilepton production processes in the Drell--Yan continuum region and at the
$Z$-pole. We also calculate the $\sin(2\phi-\phi_S)$, $\sin(2\phi+\phi_S)$ and
$\sin2\phi$ azimuthal asymmetries of $pp$ and $pd$ dilepton production
processes with the target proton and deuteron longitudinally or transversally
polarized in the Drell--Yan continuum region and around $Z$ resonances region.
We conclude that it is feasible to measure these azimuthal asymmetries,
consequently the three-dimensional or transverse momentum dependent parton
distribution functions (3dPDFs or TMDs), at this new AFTER facility.
