On sums of sparse prime subsets

TL;DR

This paper investigates the sumsets of sparse subsets of primes, establishing lower bounds on their sizes based on the subset density, which advances understanding of additive properties of prime subsets.

Contribution

It provides new lower bounds on the size of sumsets of sparse prime subsets, extending previous results to more general sparse conditions.

Findings

Sumsets of prime subsets are large under certain sparsity conditions.

Lower bounds depend on the density parameter and logarithmic factors.

Results apply to subsets with density as low as inverse powers of log x.

Abstract

For arbitrary $c_0>0$, if $A$ is a subset of the primes less than $x$ with cardinality $\delta x (\log x)^{-1}$ with $\delta\geq (\log x)^{-c_0}$, then there exists a positive constant $c$ such that the cardinality of $A+A$ is larger than $c\, \delta x (\log\log x)^{-1}$.