Vinogradov's three primes theorem in the intersection of multiple Piatetski-Shapiro sets
Xiaotian Li, Jinjiang Li, Min Zhang
TL;DR
The paper extends Vinogradov's three primes theorem to the setting where the three primes are drawn from the intersection of multiple Piatetski--Shapiro sequences. Using a refined circle-method framework, combined with sharp exponential-sum estimates for phases with fractional powers and a Heath–Brown type expansion to handle primes, the authors establish an asymptotic formula with a singular series as the main term. The result holds under explicit, verifiable constraints on the exponents defining the Piatetski--Shapiro sets, and it broadens the applicability of ternary Goldbach-type results to constrained prime subsets with precise error terms. This advances understanding of additive problems in sparse prime sets and provides a template for further multi-set intersection analyses in additive number theory.
Abstract
Vinogradov's three primes theorem indicates that, for every sufficiently large odd integer $N$, the equation $N=p_1+p_2+p_3$ is solvable in prime variables $p_1,p_2,p_3$. In this paper, it is proved that Vinogradov's three primes theorem still holds with three prime variables constrained in the intersection of multiple Piatetski-Shapiro sequences.