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Diophantine approximation with one prime, two squares of primes and one $k$-th power of a prime

Alessandro Gambini

Abstract

Let $1<k<14/5$, $λ_1,λ_2,λ_3$ and $λ_4$ be non-zero real numbers, not all of the same sign such that $λ_1/λ_2$ is irrational and let $ω$ be a real number. We prove that the inequality $|λ_1p_1+λ_2p_2^2+λ_3p_3^2+λ_4p_4^k-ω|\le (\max_j p_j)^{-\frac{14-5k}{28k}+\varepsilon}$ has infinitely many solutions in prime variables $p_1,p_2,p_3,p_4$ for any $\varepsilon>0$.

Diophantine approximation with one prime, two squares of primes and one $k$-th power of a prime

Abstract

Let , and be non-zero real numbers, not all of the same sign such that is irrational and let be a real number. We prove that the inequality has infinitely many solutions in prime variables for any .

Paper Structure

This paper contains 10 sections, 11 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Outline of the proof
  3. Setting the problem
  4. Lemmas
  5. The major arc
  6. Main Term: lower bound for $J_1$
  7. Bound for $J_4$
  8. Bound for $J_5$
  9. The trivial arc
  10. The minor arc

Key Result

Theorem 1

Assume that $1<k<14/5$, $\lambda_1,\lambda_2,\lambda_3$ and $\lambda_4$ be non-zero real numbers, not all of the same sign, that $\lambda_1/\lambda_2$ is irrational and let $\omega$ be a real number. The inequality has infinitely many solutions in prime variables $p_1,p_2,p_3,p_4$ for any $\varepsilon>0$.

Theorems & Definitions (11)

  • Theorem 1
  • Lemma 1: Languasco-Zaccagnini2016, Theorem 1
  • Lemma 2: Languasco-Zaccagnini2016, Theorem 2
  • Lemma 3: GLZ, Lemma 3
  • Lemma 4: GLZ, Lemma 4
  • Lemma 5: GLZ, Lemma 10
  • Lemma 6
  • Lemma 7: Vaughan vaughan1997hardy, Theorem 3.1
  • Corollary 1: Liu-Sun liu2013diophantine, Corollary 2.7
  • Lemma 8: Wang-Yao wang-yao, Lemma 1
  • ...and 1 more