Small solutions to linear forms in primes

TL;DR

The paper proves that a positive proportion of primitive 4-variable linear forms in primes have local solvability and, for most such forms, admit a small prime solution to $a_1p_1+a_2p_2+a_3p_3+a_4p_4=0$ with $m(\mathbf{a})^3$ bounded by $|\mathbf{a}|(\log|\mathbf{a}|)^4(\log\log|\mathbf{a}|)$. The authors combine a circle-method–type local model with a geometry-of-numbers framework to control second moments via a variance bound, and they link the local counting function to singular series/integral factors. They establish lower bounds for both the singular series and singular integral on average, and prove a positive density result for locally solvable tuples, including an explicit subset with asymptotically $CA^4$ growth. Together, these components show that small prime solutions are not rare on average and quantify the structure of locally solvable linear forms in primes. The work advances understanding of prime-solvable linear equations in several variables through a synthesis of analytic number theory and lattice-point geometry with averaging over coefficients.

Abstract

We show that a positive proportion of linear forms in four variables admit a solution in the primes that is as small as one would heuristically expect. Out of the linear forms that satisfy certain local solvability conditions, almost all admit small prime solutions.

Theorems & Definitions (50)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Proposition 2.1
• Corollary 2.2
• Proposition 2.3
• proof : Proof of Theorem \ref{['theorem:main_theorem']}
• Definition 3.1
• Lemma 3.2
• Lemma 3.3