The power benefits of using a colder, denser induction charge is proven.
The principle is simple, when the engine swallows air from a single cylinder cycle it does it by volume, not by mass. In real terms this means the warmer the air the less mass it swallows. Combustion is by mass, not by volume so you can burn more air-fuel mix in a single cycle when the air is colder.
Turbo charged engines have the problem in reverse. The turbo heats the induction charge up substantially which in turn significantly reduces the air-fuel potential. An intercooler acts to reverse the problem but can only go so far.
Cryogenic systems exist that can cool the charge below ambient air temperature but they are not suitable for constant operation so you don't tend to find them outside of drag racing.
What it means to an standard X1/9 is that you can gain a bit of power simply by opening up one of the side vents and ducting it straight into the air filter. The important detail is that it doesn't draw the air from directly inside the engine bay which tends to be a pretty warm place...