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Diophantine approximation with mixed powers of Piatetski-Shapiro primes

S. I. Dimitrov

Abstract

Let $[\,\cdot\,]$ denote the floor function. In this paper, we show that whenever $η$ is real and the constants $λ_i$ satisfy some necessary conditions, then for any fixed $\frac{63}{64}<γ<1$ and $θ>0$, there exist infinitely many prime triples $p_1,\, p_2,\, p_3$ satisfying the inequality \begin{equation*} |λ_1p_1 + λ_2p_2 + λ_3p^2_3+η|<\big(\max \{p_1, p_2, p^2_3\}\big)^{{\frac{63-64γ}{52}}+θ} \end{equation*} and such that $p_i=[n_i^{1/γ}]$, $i=1,\,2,\,3$.

Diophantine approximation with mixed powers of Piatetski-Shapiro primes

Abstract

Let denote the floor function. In this paper, we show that whenever is real and the constants satisfy some necessary conditions, then for any fixed and , there exist infinitely many prime triples satisfying the inequality \begin{equation*} |λ_1p_1 + λ_2p_2 + λ_3p^2_3+η|<\big(\max \{p_1, p_2, p^2_3\}\big)^{{\frac{63-64γ}{52}}+θ} \end{equation*} and such that , .

Paper Structure

This paper contains 8 sections, 10 theorems, 48 equations.

Table of Contents

  1. Introduction and statement of the result
  2. Notations
  3. Preliminary lemmas
  4. Beginning of the proof
  5. Lower bound for $\mathbf{\Gamma_1(X)}$
  6. Upper bound for $\mathbf{\Gamma_2(X)}$
  7. Upper bound for $\mathbf{\Gamma_3(X)}$
  8. Proof of the Theorem

Key Result

Theorem 1

Suppose that $\lambda_1,\lambda_2,\lambda_3$ are nonzero real numbers, not all of the same sign, that $\lambda_1/\lambda_2$ is irrational, and that $\eta$ is real. Let $\theta>0$ and $\gamma$ be fixed with $\frac{63}{64}<\gamma<1$. Then there exist infinitely many ordered triples of Piatetski-Shapir

Theorems & Definitions (19)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 9 more