  Unlike crystalline solids, liquids do not exhibit long-range order. Their
atoms undergo frequent structural rearrangements, resulting in the
long-wavelength dynamics of shear fluctuations in liquids being diffusive,
rather than propagating waves, as observed in crystals. When considering
shorter time and length scales, molecular dynamics simulations and theoretical
propositions suggest that collective shear excitations in liquids display a gap
in wave-vector space, referred to as the $k$-gap. Above this gap, solid-like
transverse waves re-emerge. However, direct experimental verification of this
phenomenon in classical liquids remains elusive, with the only documented
evidence from studies in two-dimensional dusty plasmas. Active granular systems
provide a novel platform for exploring the emergence of collective dynamics and
showcasing a rich interplay of complex phases and phenomena. Our study focuses
on bi-disperse active Brownian vibrators. Through measurements of the pair
correlation functions, mean square displacements, velocity auto-correlation
functions, vibrational density of states, and a detailed analysis of
microscopic atomic motion, we demonstrate that this active system exhibits both
gas-like and liquid-like phases, depending on the packing fraction, despite
pure hard-disk-like repulsive interactions. Notably, within the granular
liquid-like phase, we experimentally validate the existence of a $k$-gap in the
dispersion of transverse excitations. This gap becomes more significant with a
decrease in packing fraction and disappears into the gas phase, aligning with
theoretical expectations. Our results offer a direct experimental confirmation
of the $k$-gap phenomenon, extending its relevance beyond classical thermal
liquids to active granular systems, and reveal the existence of intriguing
similarities between the physics of active granular matter and supercritical
fluids.
