On the density of integers of the form 2k + p in arithmetic progressions

Consider all the arithmetic progressions of odd numbers, no term of which is of the form 2k + p, where k is a positive integer and p is an odd prime. Erd.s ever asked whether all these progressions can be obtained from covering congruences. In this paper, we characterize all arithmetic progressions in which there are positive proportion natural numbers that can be expressed in the form 2k +p, and give a quantitative form of Romanoff.s theorem on arithmetic progressions. As a corollary, we prove that the answer to the above Erd.s problem is affirmative.