The Exceptional Set in Goldbach's Problem with Almost Twin Primes

TL;DR

This paper establishes a power-saving bound for the exceptional set in a variant of Goldbach's problem involving sums of almost twin primes, using advanced sieve techniques and circle method generalizations.

Contribution

It introduces a new transference principle for sieves and combines multiple advanced estimates to improve bounds on the exceptional set in Goldbach's problem with almost twin primes.

Findings

Power-saving bound for the exceptional set established

New sieve transference principle developed

Enhanced circle method with sieve weights applied

Abstract

We consider the exceptional set in the binary Goldbach problem for sums of two almost twin primes. Our main result is a power-saving bound for the exceptional set in the problem of representing $m=p_1+p_2$ where $p_1+2$ has at most $2$ prime divisors and $p_2+2$ has at most $3$ prime divisors. There are three main ingredients in the proof: a new transference principle like approach for sieves, a combination of the level of distribution estimates of Bombieri--Friedlander--Iwaniec and Maynard with ideas of Drappeau to produce power savings, and a generalisation of the circle method arguments of Montgomery and Vaughan that incorporates sieve weights.