  Scalar field theories with a shift symmetry come equipped with the
$K$-mouflage (or kinetic screening) mechanism that suppresses the scalar
interaction between massive objects below a certain distance, the screening
radius. In this work, we study the linear response of the scalar field
distribution around a screened (point-like) object subject to a long range
external scalar field perturbation for the Dirac-Born-Infeld theory. We find
that, for regular boundary conditions at the position of the particle, some
multipoles have vanishing response for a lacunar series of the multipole order
$\ell$ for any dimension. Some multipoles also exhibit a logarithmic running
when the number of spatial dimensions is even. We construct a ladder operator
structure, with its associated ladder symmetries, formed by two sets of ladders
that are related to the properties of the linear response and the existence of
conserved charges. Our results exhibit a remarkable resemblance with the Love
numbers properties of black holes in General Relativity, although some
intriguing differences subsist.
