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Large value estimates in number theory, harmonic analysis, and computer science

Larry Guth

TL;DR

This survey synthesizes large value problems across analytic number theory, harmonic analysis, and computer science, focusing on Dirichlet polynomials and the operator-norm/power-method framework, augmented by Schatten-tensor and decoupling techniques. It articulates the central conjectures for Dirichlet polynomials, explains evidence from orthogonality, random-matrix heuristics, and wave-packet methods, and discusses computational barriers (low-degree testing and sum-of-squares) that may limit polynomial-time proofs. By contrasting structured matrices like $M_{Dir}$ with random models and detailing barriers linked to Kakeya geometry, the work clarifies where classical methods succeed, where they fail, and where new structural insights are needed. The paper highlights the role of wave packets and restriction theory in generating sharp bounds in certain Fourier-analytic settings, while revealing the current limitations in the Dirichlet-polynomial regime, especially when $\sigma\le 3/4$. Overall, it maps a landscape of techniques, conjectures, and barriers that guide future progress on large value problems across disciplines.

Abstract

We survey large value problems, including the large value problem for Dirichlet polynomials, the restriction problem, and problems from computer science. We describe known techniques and open problems, drawing on perspectives from all three fields.

Large value estimates in number theory, harmonic analysis, and computer science

TL;DR

This survey synthesizes large value problems across analytic number theory, harmonic analysis, and computer science, focusing on Dirichlet polynomials and the operator-norm/power-method framework, augmented by Schatten-tensor and decoupling techniques. It articulates the central conjectures for Dirichlet polynomials, explains evidence from orthogonality, random-matrix heuristics, and wave-packet methods, and discusses computational barriers (low-degree testing and sum-of-squares) that may limit polynomial-time proofs. By contrasting structured matrices like with random models and detailing barriers linked to Kakeya geometry, the work clarifies where classical methods succeed, where they fail, and where new structural insights are needed. The paper highlights the role of wave packets and restriction theory in generating sharp bounds in certain Fourier-analytic settings, while revealing the current limitations in the Dirichlet-polynomial regime, especially when . Overall, it maps a landscape of techniques, conjectures, and barriers that guide future progress on large value problems across disciplines.

Abstract

We survey large value problems, including the large value problem for Dirichlet polynomials, the restriction problem, and problems from computer science. We describe known techniques and open problems, drawing on perspectives from all three fields.

Paper Structure

This paper contains 31 sections, 12 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Large value problems in different parts of math
  3. Quantitative linear algebra
  4. The large value problem for Dirichlet polynomials
  5. Large value estimates in Fourier analysis and PDE
  6. Large value estimates in statistics and computer science
  7. Examples and conjectures
  8. Examples
  9. Random matrices
  10. Examples based on approximate geometric series
  11. The main conjectures about Dirichlet polynomials
  12. Evidence for the main conjectures
  13. Large value estimates from the computational point of view
  14. Classical methods for studying large value problems
  15. Orthogonality / operator norm method
  16. ...and 16 more sections

Key Result

Proposition 3.1

Suppose that $1/2 < \sigma \le 1$. Suppose that $U \subset \{1, ..., T \}$ is any subset with $|U| \sim N^{2 - 2 \sigma}$. Then we can choose a vector $b$ so that $|b_n| \lessapprox 1$ and so that $|(Mb)_t| \gtrsim N^\sigma$ for $t \in W$, where $W \subset U$ with $|W| \sim N^{2 - 2 \sigma}$.

Theorems & Definitions (26)

  • Definition 2.1
  • Proposition 3.1
  • Proposition 3.2
  • Conjecture 3.3: Montgomery's large value conjecture, cf M2 page 142
  • Conjecture 3.4: Montgomery's $\ell^q$ conjecture
  • Proposition 5.1
  • proof : Proof idea
  • Proposition 5.2
  • Proposition 5.3
  • proof
  • ...and 16 more