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A plane contact problem of elasticity theory on the interaction of an elastic half-space and a rigid base is formulated with due regard for the presence of a gas in the gap. It is assumed that the gas is ideal and compressible and that its state can be described by the Clayperon–Mendeleyev equation. The mathematical model of the contact behavior of the system as it is loaded takes account of the transforming height and width of the intercontact gap, as well as of the changing gas pressure in the gap. The solution to the problem is presented in terms of a gap-height function, which can be obtained from a certain singular integral equation. The width of the gap is derived from an additional condition that ensures boundedness of the contact pressure. The problem is solved analytically for one form of the gap. The graphs presented illustrate the influence of the gas and its mass on the geometric parameters of the gap contact pressure.