Premium
A note on additive properties of dense subsets of sifted sequences
Author(s) -
Ramaré O.
Publication year - 2013
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bds100
Subject(s) - mathematics , sequence (biology) , argument (complex analysis) , combinatorics , discrete mathematics , natural density , arithmetic , natural number , biochemistry , chemistry , genetics , biology
In this paper, we show that if A is a subset of the primes with positive lower density 1/ k , then A + A must have positive lower density at least C 1 /[ k log log(3 k )] in the natural numbers. Our argument uses the techniques developed by the author and I. Ruzsa in their work on additive properties of dense subsequences of sufficiently sifted sequences. The result is optimal and improves on recent work of K. Chipeniuk and M. Hamel. We continue by proving several similar results, by successively replacing the sequence of primes by the sequence of sums of two squares, by the sequence of those integers n that are such that n and n + 1 are both a sum of two squares and finally by the sequence of primes p that are such that p + 1 is a sum of two squares. The second part of this paper contains a heuristical argument that leads to several conjectures concerning the existence of arithmetic progressions of arbitrary length within these sequences. We conclude with some conjectures belonging to the Ramsey part of additive number theory.