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Effective Results in The Metric Theory of Quantitative Diophantine Approximation

Ying Wai Lee, Andrew Scoones

Abstract

Many results related to quantitative problems in the metric theory of Diophantine approximation are asymptotic, such as the number of rational solutions to certain inequalities grows with the same rate almost everywhere modulo an asymptotic error term. The error term incorporates an implicit constant that varies from one point to another. This means that applications of these results does not give concrete bounds when applied to, say a finite sum, or when applied to counting the number of solutions up to a finite point for a given inequality. This paper addresses this problem and makes the tools and their results effective, by making the implicit constant explicit outside of an exceptional subset of Lebesgue measure at most $δ>0$, an arbitrarily small constant chosen in advance. We deduce from this the fully effective results for Schmidt's Theorem, quantitative Koukoulopoulos-Maynard Theorem and quantitative results on $M_{0}$-sets; we also provide effective results regarding statistics of normal numbers and strong law of large numbers.

Effective Results in The Metric Theory of Quantitative Diophantine Approximation

Abstract

Many results related to quantitative problems in the metric theory of Diophantine approximation are asymptotic, such as the number of rational solutions to certain inequalities grows with the same rate almost everywhere modulo an asymptotic error term. The error term incorporates an implicit constant that varies from one point to another. This means that applications of these results does not give concrete bounds when applied to, say a finite sum, or when applied to counting the number of solutions up to a finite point for a given inequality. This paper addresses this problem and makes the tools and their results effective, by making the implicit constant explicit outside of an exceptional subset of Lebesgue measure at most , an arbitrarily small constant chosen in advance. We deduce from this the fully effective results for Schmidt's Theorem, quantitative Koukoulopoulos-Maynard Theorem and quantitative results on -sets; we also provide effective results regarding statistics of normal numbers and strong law of large numbers.
Paper Structure (17 sections, 36 theorems, 343 equations)

This paper contains 17 sections, 36 theorems, 343 equations.

Table of Contents

  1. Introduction
  2. Probabilistic Results and Proof of an Effective Schmidt's Theorem
  3. Probabilistic Results
  4. Proof of Theorem \ref{['EST']} modulo Theorem \ref{['thm3']}
  5. Further Applications
  6. Quantitative Koukoulopoulos-Maynard Theorem
  7. Inhomogeneous Diophantine Approximation on $M_{0}$-sets
  8. Normal Numbers
  9. Strong Law of Large Numbers
  10. Proof of the Probabilistic Results
  11. Proof of Theorem \ref{['thm2']}
  12. Proof of Theorem \ref{['thm3']}
  13. Proof of Lemma \ref{['Aist general thm']}
  14. Proof of Theorem \ref{['effective v3']}
  15. Appendix
  16. ...and 2 more sections

Key Result

Theorem 1

For any $x \in \mathbb{R}$ and $N \in \mathbb{N}$, there exist $p,\,q \in \mathbb{Z}$ such that with $1 \leq q \leq N$.

Theorems & Definitions (59)

  • Theorem : Dirichlet's Theorem, 1842
  • Theorem
  • Theorem : Hurwitz's Theorem, 1891
  • Theorem : Khintchine, 1924
  • Theorem : Koukoulopoulos-Maynard, 2019
  • Theorem : Schmidt, 1960
  • Theorem 1: Effective Schmidt's Theorem
  • Theorem 2: Effective Version of Lemma 1.4 from harman1998metric
  • Theorem 3: Effective Version of Lemma 1.5 from harman1998metric
  • Lemma 1: Effective Lemma 4.1 from harman1998metric
  • ...and 49 more