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Romanoff theorem in a sparse set

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Abstract

Let A be any subset of positive integers, and P the set of all positive primes. Two of our results are: (a) the number of positive integers which are less than x and can be represented as 2k + p (resp. p − 2k) with k ∈ A and p ∈ P is more than 0.03A(log x/ log 2)π(x) for all sufficiently large x; (b) the number of positive integers which are less than x and can be represented as 2q + p with p, q ∈ P is (1 + o(1)π(log x/ log 2)π(x). Four related open problems and one conjecture are posed.

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Authors and Affiliations

  1. School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210046, China

    Yong-Gao Chen

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  1. Yong-Gao Chen
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Correspondence to Yong-Gao Chen.

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Dedicated to Professor Wang Yuan on the Occasion of his 80th Birthday

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Chen, YG. Romanoff theorem in a sparse set. Sci. China Math. 53, 2195–2202 (2010). https://doi.org/10.1007/s11425-010-3084-x

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  • DOI: https://doi.org/10.1007/s11425-010-3084-x

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