  We analyze the phase diagram of the exact ground state (GS) of spin-$s$
chains with ferromagnetic $XXZ$ couplings under $n$-alternating field
configurations, i.e, sparse alternating fields having nodes at $n-1$ contiguous
sites. It is shown that such systems can exhibit a non-trivial magnetic
behavior, which can differ significantly from that of the standard ($n=1$)
alternating case and enable mechanisms for controlling their magnetic and
entanglement properties. The boundary in field space of the fully aligned phase
can be determined analytically $\forall\,n$, and shows that it becomes
reachable only above a threshold value of the coupling anisotropy $J_z/J$,
which depends on $n$ but is independent of the system size. Below this value
the maximum attainable magnetization becomes much smaller. We then show that
the GS can exhibit significant magnetization plateaus, persistent for large
systems, at which the magnetization per site $m$ obeys the quantization rule
$2n(s-m)=integer$, consistent with the Oshikawa, Yamanaka and Affleck (OYA)
criterion. We also identify the emergence of field induced spin polymerization,
which explains the presence of such plateaus. Entanglement and field induced
frustration effects are also analyzed.
