  We investigate the localization pattern of interacting impurities, which are
trapped in a lattice potential and couple to a Bose gas. For small interspecies
interaction strengths, the impurities populate the energetically lowest Bloch
state or localize separately in different wells with one extra particle being
delocalized over all the wells, depending on the lattice depth. In contrast,
for large interspecies interaction strengths we find that due to the fractional
filling of the lattice and the competition of the repulsive contact interaction
between the impurities and the attractive interaction mediated by the Bose gas,
the impurities localize either pairwise or completely in a single well. Tuning
the lattice depth, the interspecies and intraspecies interaction strength
correspondingly allows for a systematic control and engineering of the two
localization patterns. The sharpness of the crossover between the two states as
well as the broad region of their existence supports the robustness of the
engineering. Moreover, we are able to manipulate the ground state's degeneracy
in form of triplets, doublets and singlets by implementing different boundary
conditions, such as periodic and hard wall boundary conditions.
