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

Consider all the arithmetic progressions of odd numbers, no term of which is of the form 2 ^k + p, where k is a positive integer and p is an odd prime. Erdős 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 2 ^k +p, and give a quantitative form of Romanoff’s theorem on arithmetic progressions. As a corollary, we prove that the answer to the above Erdős problem is affirmative.