A density theorem for higher-order sums of prime numbers

Abstract Let P be a subset of the primes of lower density strictly larger than $$\frac{1}{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:math> . Then, every sufficiently large even integer is a sum of four primes from the set P . We establish similar results for k -summands, with $$k\geqslant 4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> , and for $$k \geqslant 4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> distinct subsets of primes. This extends the work of H. Li, H. Pan, as well as X. Shao on sums of three primes, and A. Alsteri and X. Shao on sums of two primes. The primary new contributions come from elementary combinatorial lemmas.