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Diophantine approximation with integers having no large prime factors

Kunjakanan Nath, Habibur Rahaman

Abstract

Given any irrational number $α$ and a real number $κ>0$, we show that for any $θ<θ_0(y)$, there are infinitely many $y$-smooth (friable) numbers $n$ such that $$\|nα\|<n^{-θ},$$ where $θ_0(y)=6/17$ if $\exp((\log\log n)^{2+κ})\leq y\leq n^{o(1)}$ and $θ_0(y)=20/59$ if $n^{o(1)}<y\leq n$. Here $\|x\|$ denotes the distance from $x$ to the nearest integer. Our proof is based on the dispersion method together with arithmetic inputs coming from the average bounds for Kloosterman sums over smooth numbers.

Diophantine approximation with integers having no large prime factors

Abstract

Given any irrational number and a real number , we show that for any , there are infinitely many -smooth (friable) numbers such that where if and if . Here denotes the distance from to the nearest integer. Our proof is based on the dispersion method together with arithmetic inputs coming from the average bounds for Kloosterman sums over smooth numbers.
Paper Structure (27 sections, 27 theorems, 229 equations)

This paper contains 27 sections, 27 theorems, 229 equations.

Table of Contents

  1. Introduction
  2. Motivation and previous results
  3. Main result
  4. Notation
  5. Plan of the paper
  6. Set-up and proof of Theorem \ref{['main_thm']}
  7. Proof sketch of Theorem \ref{['main_thm2']}
  8. Preliminaries
  9. Auxiliary results on smooth numbers
  10. Exponential sum bounds
  11. Elementary bounds
  12. Estimates for averages of Kloosterman sums
  13. Type I sums
  14. Type I sums via Vinogradov's methods
  15. Type I sums via geometry of numbers
  16. ...and 12 more sections

Key Result

Theorem 1

Let $\alpha$ be a given irrational number and let $\kappa>0$. Then, for any $\theta<\theta_0(Y)$, there are infinitely many $Y$-smooth numbers $n$ such that where Here we use the standard convention that the above $o(1)$ term tends to $0$ as $n$ tends to infinity.

Theorems & Definitions (61)

  • Theorem 1
  • Remark 1.1
  • Definition 2.1
  • Remark 2.1
  • Theorem 2
  • proof : Proof of Theorem \ref{['main_thm']} assuming Theorem \ref{['main_thm2']}
  • Lemma 3.1
  • proof
  • Remark 3.1
  • Lemma 3.2
  • ...and 51 more