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Sharpness of the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem in all dimensions

Robert Fraser, Kyle Hambrook, Donggeun Ryou

TL;DR

The paper establishes the sharpness of the exponent p_*(a,b,d) in the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem for all dimensions d and all 0<a,b<d. It does so by constructing a deterministic Salem-type measure μ supported on a limsup set E(K,B,τ) derived from algebraic number theory and by demonstrating that Fourier extension estimates fail below a specific p determined by a refined multi-parameter threshold p(τ,ρ,q,d). The construction hinges on intermediate measures, inversion to lattices via ideals, precise Fourier decay, and a separation-based regularity analysis, culminating in a fundamentally sharp obstruction to restriction in this fractal setting. The results deliver a complete resolution of the sharpness problem in this regime and provide deterministic Salem-type counterexamples with controlled Hausdorff dimension, advancing the understanding of Fourier restriction on fractal measures and the role of number-theoretic structure in restriction phenomena.

Abstract

We prove the optimality of the exponent in the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem in all dimensions $d$ and the full parameter range $0 < a,b < d$. Our construction is deterministic and also yields Salem sets.

Sharpness of the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem in all dimensions

TL;DR

The paper establishes the sharpness of the exponent p_*(a,b,d) in the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem for all dimensions d and all 0<a,b<d. It does so by constructing a deterministic Salem-type measure μ supported on a limsup set E(K,B,τ) derived from algebraic number theory and by demonstrating that Fourier extension estimates fail below a specific p determined by a refined multi-parameter threshold p(τ,ρ,q,d). The construction hinges on intermediate measures, inversion to lattices via ideals, precise Fourier decay, and a separation-based regularity analysis, culminating in a fundamentally sharp obstruction to restriction in this fractal setting. The results deliver a complete resolution of the sharpness problem in this regime and provide deterministic Salem-type counterexamples with controlled Hausdorff dimension, advancing the understanding of Fourier restriction on fractal measures and the role of number-theoretic structure in restriction phenomena.

Abstract

We prove the optimality of the exponent in the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem in all dimensions and the full parameter range . Our construction is deterministic and also yields Salem sets.

Paper Structure

This paper contains 28 sections, 42 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Main Result
  3. Deterministic Salem Sets and a Variant of E(k,B,tau.
  4. Notation
  5. Organization of The Paper
  6. Motivation and Discussion
  7. Restriction Theorems
  8. Sharpness of Theorem \ref{['frac ST']}.
  9. Key Ideas in the Proof of Theorem \ref{['mainthm3']}
  10. Algebraic Number Theory
  11. Intermediate Functions
  12. The Functions phi and Phi
  13. The collection of ideals Q(M)
  14. The Functions Fk
  15. Intermediate Measures
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $0 < a, b < d$, with $d$ an integer. Define $p_{*}(a,b,d) = (4d-4a+2b)/b.$ Suppose $\mu$ is a Borel measure on $\mathbb{R}^d$ with the following properties: Then, for each $p \geq p_{*}(a,b,d)$, the following holds:

Theorems & Definitions (80)

  • Theorem 1.1: Mockenhaupt-Mitsis-Bak-Seeger
  • Theorem 1.2: Main Theorem
  • Remark 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 70 more