The growth of Co on Cu͑111͒ at coverage 0.03 monolayer is studied by kinetic Monte Carlo simulations performed above temperature of dimer formation ͑20-21 K͒ in order to find a nanostructure composed primarily of dimers as an analog of the dilute nanostructure formed of monomers. Both of the nanostructures are stabilized by a long-range surface-state-mediated interaction between adatoms. In order to obtain the largest possible fraction of dimers the simulation procedure consists in deposition of adatoms at low temperature ͑7 K͒ and simulated annealing at higher temperatures. After 60-200 s of annealing at 21-22 K, about 87% of adatoms belongs to dimers. The dimers form dilute islands with a local hexagonal order and an average nearest-neighbor distance of 12.0͑5͒ Å. The nanostructure in the dimer state is stable up to 200 K.