We prove that if $A$ is a subset of those primes which are congruent to $1 \pmod {3}$ such that the relative density of $A$ in this residue class is larger than $\frac{1}{2},$ then every sufficiently large odd integer $n$ which satisfies $n \equiv 0 \pmod {3}$ can be written as a sum of three primes from $A$. Moreover the threshold of $\frac{1}{2}$ for the relative density is best possible.
Nous prouvons que si $A$ est un sous-ensemble des nombres premiers qui sont congrus à $1 \pmod {3}$ tel que la densité relative de $A$ dans cette classe résiduelle soit supérieure à $\frac{1}{2}$, alors tout entier impair $n$ suffisamment grand qui satisfait $n \equiv 0 \pmod {3}$ peut être écrit comme une somme de trois nombres premiers de $A$. De plus, le seuil de $\frac{1}{2}$ pour la densité relative est optimal.
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Keywords: Goldbach-type problems, relatively dense subset, transference principle, additive combinatorics.
Ali Alsetri 1
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Ali Alsetri. Density ternary Goldbach for primes in a fixed residue class. Journal de théorie des nombres de Bordeaux, Tome 38 (2026) no. 2, pp. 451-467. doi: 10.5802/jtnb.1367
@article{JTNB_2026__38_2_451_0,
author = {Ali Alsetri},
title = {Density ternary {Goldbach} for primes in a fixed residue class},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {451--467},
year = {2026},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {38},
number = {2},
doi = {10.5802/jtnb.1367},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1367/}
}
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