Structure of sets of bounded sequences with a prescribed number of accumulation points

For each vector $x\in {\mathrm{\ell }}^{\mathrm{\infty }}$, we can define the non-empty compact set $L_x$ of accumulation points of $x$. Given an infinite subset $A$ of $\mathbb{N}\mathrm{\backslash }\{1\}$, we can therefore investigate under which conditions on $A$, the set $L(A):=\{x\in {\mathrm{\ell }}^{\mathrm{\infty }}:\mathrm{\mid }{L}_x\mathrm{\mid }\in A\}$ is lineable or even densely lineable. In particular, we show that if $L(A)$ is lineable then there exists $k\ge 1$ such that $A\cap (A-k)$ is infinite and that if $L(A)$ is densely lineable then $A\cap (A-1)$ is infinite. We end up by answering an open question on the existence of a closed non-separable subspace in which each non-zero vector has countably many accumulation points.