Primes in Sumsets

TL;DR

This paper establishes an improved upper bound on the number of pairs from two subsets of integers whose sum is prime, advancing understanding of prime distributions in sumsets.

Contribution

It provides a tighter upper bound for prime-sum pairs in large subsets, improving previous results by Balog, Rivat, and Sárközy.

Findings

Derived a new upper bound for prime-sum pairs

Enhanced previous bounds by Balog, Rivat, and Sárközy

Applicable for large subsets with product size significantly exceeding N^2/(log N)^2

Abstract

We obtain an upper bound for the number of pairs $ (a,b) \in {A\times B} $ such that $ a+b $ is a prime number, where $ A, B \subseteq \{1,...,N \}$ with $|A||B| \, \gg \frac{N^2}{(\log {N})^2}$, $\, N \geq 1$ an integer. This improves on a bound given by Balog, Rivat and S\'ark\"ozy.