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Level set estimates for the periodic Schrödinger maximal function on $\mathbb{T}^1$

Ciprian Demeter

TL;DR

The paper addresses level set estimates for the periodic Schrödinger maximal function on $\,\mathbb{T}^1$, proving essentially sharp $L^4$ bounds in a specific cutoff range and linking the problem to arithmetic-combinatorial counting. It translates the problem into a graph-theoretic counting framework via a TT* reduction and a quadratic kernel $\mathcal{K}$, then develops a detailed partitioning by gcd data into parameters $D,P,F$ and a fork structure to control incidences. The main contributions are bounds of the form $R \lesssim N/M^4$ for the number of significant sample points, valid in the regime $M \gtrapprox N^{1/10}$ (with improvements under various subcases), along with a demonstration that the familiar $L^6$ bound $N^{1/3}$ can be recovered from the same framework. The work also discusses potential conditional strategies (e.g., a three-neighbor conjecture) to extend the results beyond the current barrier, indicating paths toward sharper convergence results for the Schrödinger flow on the torus. Math notation is used throughout to express the core scaling relations, kernel bounds, and counting parameters, emphasizing the arithmetic structure underlying the maximal operator estimates.

Abstract

We prove (essentially) sharp $L^4$ level set estimates for the periodic Schrödinger maximal operator in a certain range of the cut-off parameter.

Level set estimates for the periodic Schrödinger maximal function on $\mathbb{T}^1$

TL;DR

The paper addresses level set estimates for the periodic Schrödinger maximal function on , proving essentially sharp bounds in a specific cutoff range and linking the problem to arithmetic-combinatorial counting. It translates the problem into a graph-theoretic counting framework via a TT* reduction and a quadratic kernel , then develops a detailed partitioning by gcd data into parameters and a fork structure to control incidences. The main contributions are bounds of the form for the number of significant sample points, valid in the regime (with improvements under various subcases), along with a demonstration that the familiar bound can be recovered from the same framework. The work also discusses potential conditional strategies (e.g., a three-neighbor conjecture) to extend the results beyond the current barrier, indicating paths toward sharper convergence results for the Schrödinger flow on the torus. Math notation is used throughout to express the core scaling relations, kernel bounds, and counting parameters, emphasizing the arithmetic structure underlying the maximal operator estimates.

Abstract

We prove (essentially) sharp level set estimates for the periodic Schrödinger maximal operator in a certain range of the cut-off parameter.
Paper Structure (12 sections, 14 theorems, 224 equations)

This paper contains 12 sections, 14 theorems, 224 equations.

Table of Contents

  1. Introduction
  2. Initial reductions
  3. A combinatorial approach for Conjecture \ref{['c1']} in the range $M{\;\gtrapprox}\; N^{1/10}$
  4. A brief outline of the argument
  5. Pigeonholing and graph considerations
  6. Counting solutions
  7. Brief discussion of different ranges
  8. The argument in the case $FD{\;\lessapprox}\; Q$ and $P{\;\lessapprox}\; 2^l$
  9. The argument for the case $P\ggg 2^l$
  10. Proof in the case $FD{\;\gtrapprox}\; Q$
  11. Recovering the $N^{1/3}$ bound in $L^6$ and beyond
  12. A potential strategy for breaking the $M{\;\gtrapprox}\; N^{1/10}$ barrier

Key Result

Theorem 1.1

Assume $\|a_n\|_{l^2}=1$ and $N^{\frac{1}{4}+\frac{1}{10}}{\;\lessapprox}\; \lambda\le N^{\frac{1}{2}}$. Then

Theorems & Definitions (50)

  • Theorem 1.1
  • Conjecture 1.2
  • Conjecture 1.3
  • Corollary 1.4
  • Remark 2.1
  • Lemma 2.2
  • Remark 2.3
  • Conjecture 2.4
  • Proposition 2.5
  • Definition 3.1
  • ...and 40 more