A Diophantine inequality involving different powers of primes of the form {\boldmath$[n^c]$}

TL;DR

The paper proves that for k=4, there are infinitely many prime triples $p_1,p_2,p_3$ of Piatetski–Shapiro type with $p_i=[n_i^{1/\gamma}]$ and $\frac{219}{220}<\gamma<1$ satisfying $|\lambda_1 p_1+\lambda_2 p_2+\lambda_3 p_3^4+\eta|< (\max\{p_1,p_2,p_3^4\})^{(219-220\gamma)/208+\theta}$, where $\lambda_1/\lambda_2$ is irrational and $\eta$ is real. The approach uses the circle method, defining exponential sums $S_k(t)$, $\Sigma(t)$, $U(t)$, and $\Omega(t)$, and decomposing the main integral into three arcs $\Gamma_1,\Gamma_2,\Gamma_3$. A positive lower bound is established on the main arc $|t|<\Delta$, while precise upper bounds on the middle and outer arcs render the total contribution positive for infinitely many $X$, yielding infinitely many solutions. The results extend prior work on Diophantine inequalities with three primes to the case where one prime is raised to the fourth power and restricted to Piatetski–Shapiro primes with $\gamma$ in $(\tfrac{219}{220},1)$.

Abstract

Let $[\, x\,]$ denote the integer part of a real number $x$. Assume that $λ_1,λ_2,λ_3$ are nonzero real numbers, not all of the same sign, that $λ_1/λ_2$ is irrational, and that $η$ is real. Let $\frac{219}{220}<γ<1$ and $θ>0$. We establish that, there exist infinitely many triples of primes $p_1,\, p_2,\, p_3$ satisfying the inequality \begin{equation*} |λ_1p_1 + λ_2p_2 + λ_3p^4_3+η|<\big(\max \{p_1, p_2, p^4_3\}\big)^{\frac{219-220γ}{208}+θ} \end{equation*} and such that $p_i=[n_i^{1/γ}]$, $i=1,\,2,\,3$.

Theorems & Definitions (22)

• Theorem 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5