On a system of two Diophantine inequalities with five prime variables
Min Zhang, Jinjiang Li, Linji Long, Yuhan Yang
TL;DR
The paper advances the Waring–Goldbach–type theory for representations of two real Diophantine inequalities with five prime variables by extending the admissible range of exponents to $1<d<c<39/37$ and refining the error terms. The authors convert the problem into assessing a two-variable exponential sum over primes, decompose primes via Heath–Brown's identity, and perform a delicate major/minor arc analysis across a three-region partition, deriving a sharp bound on the critical sum in the intermediate region. A key technical achievement is the bound $S(x,y) \ll X^{34/37}(\log X)^{205}$ on the critical region $\Omega_2$, which ensures the auxiliary majorant $\mathscr{B}$ grows unbounded for large $X$, yielding solvability in primes for the stated system with explicit error terms $\varepsilon_1(N_1)$ and $\varepsilon_2(N_2)$. The result improves previous work by Zhai and Tolev, pushing the known thresholds closer to optimal ranges and strengthening the Diophantine approximation framework for prime variables.
Abstract
Suppose that $c,d,α,β$ are real numbers satisfying the inequalities $1<d<c<39/37$ and $1<α<β<5^{1-d/c}$. In this paper, it is proved that, for sufficiently large real numbers $N_1$ and $N_2$ subject to $α\leqslant N_2/N_1^{d/c}\leqslantβ$, the following Diophantine inequalities system \begin{equation*} \begin{cases} \big|p_1^c+p_2^c+p_3^c+p_4^c+p_5^c-N_1\big|<\varepsilon_1(N_1) \\ \big|p_1^d+p_2^d+p_3^d+p_4^d+p_5^d-N_2\big|<\varepsilon_2(N_2) \end{cases} \end{equation*} is solvable in prime variables $p_1,p_2,p_3,p_4,p_5$, where \begin{equation*} \begin{cases} \varepsilon_1(N_1)=N_1^{-(1/c)(39/37-c)}(\log N_1)^{201}, \\ \varepsilon_2(N_2)=N_2^{-(1/d)(39/37-d)}(\log N_2)^{201}. \end{cases} \end{equation*} This result constitutes an improvement upon a series of previous results of Zhai [14] and Tolev [12].