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Fourier decay of measures supported on sets of numbers with consecutive partial quotients belonging to a given set

Robert Fraser

TL;DR

This work advances the Fourier-analytic study of Diophantine sets by proving that sets defined via long blocks of partial quotients, specified by a general assignment S, support Rajchman measures. The authors construct a measure λ on infinite integer sequences whose continued-fraction pushforward is supported on E(S,∞) and exhibits explicit Fourier decay, built from a Kaufman-type typical/exceptional block framework with carefully chosen scale parameters i_n and r_n. The analysis blends Kaufman’s oscillatory-integral machinery with a novel sparse-exception scheme to achieve decay bounds on hat{g^{ ext{#}}λ} and yields quantitative decay rates in terms of ξ and the construction data. The results generalize prior Fourier-decay constructions for well- and badly-approximable sets and provide a flexible method to obtain Rajchman measures on a broad class of Diophantine-approximation sets, with concrete implications for questions about normal numbers and Fourier dimension.

Abstract

We consider measures supported on sets of irrational numbers possessing many consecutive partial quotients satisfying a condition based on the previous partial quotients. We show that under mild assumptions, such sets will always support measures whose Fourier transform decays to zero.

Fourier decay of measures supported on sets of numbers with consecutive partial quotients belonging to a given set

TL;DR

This work advances the Fourier-analytic study of Diophantine sets by proving that sets defined via long blocks of partial quotients, specified by a general assignment S, support Rajchman measures. The authors construct a measure λ on infinite integer sequences whose continued-fraction pushforward is supported on E(S,∞) and exhibits explicit Fourier decay, built from a Kaufman-type typical/exceptional block framework with carefully chosen scale parameters i_n and r_n. The analysis blends Kaufman’s oscillatory-integral machinery with a novel sparse-exception scheme to achieve decay bounds on hat{g^{ ext{#}}λ} and yields quantitative decay rates in terms of ξ and the construction data. The results generalize prior Fourier-decay constructions for well- and badly-approximable sets and provide a flexible method to obtain Rajchman measures on a broad class of Diophantine-approximation sets, with concrete implications for questions about normal numbers and Fourier dimension.

Abstract

We consider measures supported on sets of irrational numbers possessing many consecutive partial quotients satisfying a condition based on the previous partial quotients. We show that under mild assumptions, such sets will always support measures whose Fourier transform decays to zero.

Paper Structure

This paper contains 20 sections, 6 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. The theory of metric Diophantine approximation
  3. Fourier analysis and Diophantine approximation
  4. Techniques for constructing measures with rapid Fourier decay on sets in Diophantine approximation
  5. Statement of main result
  6. Notation and preliminaries
  7. Notation
  8. Review of Kaufman's construction
  9. Details of the construction
  10. An estimate on the Fourier transform of $g^{\sharp} \lambda$
  11. Proof of Lemma \ref{['ltyp_scale']}
  12. Proof of the estimate for $\widehat{g^{\sharp} \lambda_{\text{typ}}}$
  13. Oscillatory integral estimates of Kaufman
  14. Estimate of integral
  15. Estimate for $M$
  16. ...and 5 more sections

Key Result

Theorem 1.3

Let $S$ be any assignment of sets to partial quotients. Then $E(S, \infty)$ supports a Rajchman measure.

Theorems & Definitions (14)

  • Definition 1.1
  • Example 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Example 3.1
  • Lemma 4.1
  • proof : Proof of Lemma \ref{['ltyp_scale']}
  • Lemma 6.1: Integral inequality from Kaufman80
  • Lemma 6.2
  • Lemma 6.3
  • ...and 4 more