On sums of finite subsets of the primes
Genheng Zhao
TL;DR
The paper addresses lower bounds for the sumset |A+A| of sparse prime subsets A ⊂ [1,x] with |A| = αx/log x. It develops an effective, local-to-global strategy that leverages Granville's probabilistic prime framework, Brun-type sieve methods, and sharp restriction theorems to transfer a local arithmetic structure into global sumset growth. The main contribution is a two-regime lower bound, improving prior results by providing an explicit, effective bound: for α ≥ c (log x)^{−1/2} log log x, |A+A| ≥ c′ (log x)/(log log(3α^{−1})) · |A|, and for smaller α, |A+A| ≥ c′ (log x)/(log(2α^{−1})) · |A|. The approach avoids Siegel–Walfisz and yields a near-optimal dependence on α in the described ranges, with a clear path from restriction estimates on primes to additive-energy bounds. Overall, the work advances understanding of sumset structure for subsets of primes and provides tools potentially adaptable to other sparse-prime configurations.
Abstract
Let $A\subset [1,x]$ be a non-empty set of primes with $|A|= αx(\log x)^{-1}$. We prove that there exist absolute constants $c_1,c_2>0$ such that, as $x$ gets sufficiently large, we have $|A+A|\geq c_1(\log x)(\log \log 3α^{-1})^{-1}|A|$ if $α\geq c_2(\log x)^{-1/2}\log \log x$ and otherwise $|A+A|\geq c_1(\log x) (\log 2α^{-1})^{-1}|A|$.