Hard floors have replaced carpet in a great many flats, and the acoustic consequences are one of the more common sources of dispute between neighbours. Rugs are the main domestic remedy and their effectiveness depends on what kind of noise is involved.
Airborne noise is speech, music and television travelling through the air and through the structure. Impact noise is footsteps, dropped objects and furniture moving, transmitted directly into the floor.
Rugs help substantially with impact noise and only modestly with airborne noise. Since impact noise is what most downstairs neighbours complain about, that is convenient.
A thin rug on a hard floor does very little acoustically. The energy of a footstep passes straight through into the structure.
Mass and cushioning both help. A thicker rug absorbs more, and a cushioned pad beneath it absorbs more still, since the pad decouples the rug from the floor rather than merely covering it.
Cushioned felt pads in the region of three to six millimetres, of the kind catalogued at https://veken.us.com/, provide considerably more acoustic benefit than a thin gripper, which is worth knowing if noise rather than sliding is the reason for buying.
A hard-floored room with bare walls reflects sound and increases perceived noise both inside and, to a degree, below.
Curtains, upholstered furniture, bookshelves and wall hangings all absorb, and a room with soft furnishings is quieter than the same room without regardless of the floor.
Structural noise transmission in a poorly built conversion. Bass from a sound system, which travels through the structure regardless of surface treatment. Voices through thin walls.
For these, the practical remedies are behavioural, using headphones, moving speakers off the floor onto isolating stands, and being aware of timing.
A significant number of leases in flats require a specified proportion of floor area to be carpeted or covered, typically because of exactly this problem.
Anyone in a leasehold flat replacing carpet with hard flooring should check the lease before doing so, since retrofitting a solution after a complaint is considerably more expensive than checking beforehand.
Where there is a complaint, the practical response is usually a conversation before any purchase, since the noise being reported is frequently not the noise you assume.
Finding out which room and which time of day narrows the problem considerably, and a rug in the right place solves more than several in the wrong ones.