Analysis at Large: Dedicated to the Life and Work of Jean Bourgain

Preface
Contents
On the Joint Spectral Radius
	1 Introduction
	2 Extremal Norms and Barabanov Norms
	3 Explicit Bounds for Theorem 2
	4 Explicit Bounds for Bochi's Inequalities
	5 Ultrametric Complete Valued Fields
	References
The Failure of the Fractal Uncertainty Principle for the Walsh–Fourier Transform
	1 The Fractal Uncertainty Principle for the Fourier Transform
	2 The Walsh Transform
	3 The Main Result
	4 Proofs
	References
The Continuous Formulation of Shallow Neural Networks as Wasserstein-Type Gradient Flows
	1 Introduction
	2 Shallow Neural Network and Gradient Flows
		2.1 The μ Formulation
		2.2 Comparison Between the Continuous and Discrete Model
			Consistency
		2.3 The (ρ, H) Formulation
	3 PDE Formulations
		3.1 Gradient Flow in the μ Formulation
		3.2 A First PDE Approach in the (ρ, H) Formulation
			Separating Variables
			Transporting Along the Flow of ρt
		3.3 A Gradient Flow in the (ρ, H) Formulation via Propagation of Chaos
	4 Regularized Problems
		4.1 Heat Regularization
		4.2 The Porous Medium Regularization
		4.3 An Observation Without Regularization
	5 Open Questions
		5.1 Regularity and Convergence
		5.2 Multilayer Neural Networks
	References
On the Origins, Nature, and Impact of Bourgain's Discretized Sum-Product Theorem
	1 Overture
	2 Origins: Kakeya-Besicovitch Problem+
		2.1 Some Fundamental Properties of Plane Sets of Fractional Dimension
		2.2 Besicovitch Type Maximal Operators and Applications to Fourier Analysis
		2.3 Balog-Szemerédi-Gowers Lemma
		2.4 On the Dimension of Kakeya Sets and Related Maximal Inequalities
	3 Sum-Product Phenomena and the Labyrinth of the Continuum
		3.1 Freiman's Theorem and Ruzsa's Calculus
		3.2 Sum-Product Phenomena and Incidence Geometry
			Crossing Number Inequality
			Szemerédi-Trotter Theorem
			Proof of Sum-Product Inequality
		3.3 On the Erdös-Volkmann and Katz-Tao Discretized Ring Conjectures
			Erdös-Volkmann Problem
			Katz-Tao Discretized Ring Conjecture
			Labyrinth of the Continuum
		3.4 A Sum-Product Estimate in Finite Fields and Applications
	4 Discrete and Continuous Variations on the Expanding Theme
		4.1 Bemerkung über den Inhalt von Punktmengen
		4.2 Sur le problème de la mesure
		4.3 Ramanujan-Selberg Conjecture
		4.4 Expanders
		4.5 Superstrong Approximation
		4.6 On the Spectral Gap for Finitely Generated Subgroups of SU(d)
	5 Coda
	References
Cartan Covers and Doubling Bernstein-Type Inequalities on Analytic Subsets of C2
	1 Introduction
	2 Cartan's Estimate
	3 Bernstein Exponent and Number of Zeros
	4 Weierstrass' Preparation Theorem and Bernstein Exponents
	5 Resultants
	6 Refinement of the Assumption (1)
	7 Proofs of Theorems A, B, and C
	References
A Weighted Prékopa–Leindler Inequality and Sumsets withQuasicubes
	1 Introduction
	2 A Weighted Discrete Prékopa–Leindler Inequality
	3 Proof of the Main Theorem
	References
Equidistribution of Affine Random Walks on Some Nilmanifolds
	1 Introduction
		1.1 Quantitative Equidistribution
		1.2 Statement of the Main Result
		1.3 The Case of a Torus
		1.4 Consequences of the Main Theorem
		1.5 Idea of the Proof
	2 Examples
		2.1 Heisenberg Nilmanifold
		2.2 Heisenberg Nilmanifold over Number Fields
		2.3 A Non-semisimple Group of Toral Automorphisms
		2.4 A Non-example
	3 The Setup
		3.1 Hölder Functions
	4 The Main Argument
		4.1 Principal Torus Bundle
		4.2 Fourier Transform
		4.3 Essential Growth Rate
		4.4 The Cauchy-Schwarz Argument
		4.5 Proof of the Key Proposition
	5 Proof of the Main Theorems
	Appendix A: A Large Deviation Estimate
	Appendix B: The Case of a Torus
	B.1 Multiplicative Convolutions in Simple Algebras
	B.2 Fourier Decay for Linear Random Walks
	B.3 Proof of Theorems B.1 and B.2
	References
Logarithmic Quantum Dynamical Bounds for Arithmetically Defined Ergodic Schrödinger Operators with Smooth Potentials
	1 Introduction
	2 Preliminaries
		2.1 Schrödinger Operators and Transfer Matrices
		2.2 Transport Exponents
		2.3 Semialgebraic Sets
		2.4 Large Deviation Theorems
	3 Transport Exponents
	4 Semialgebraic Sets
	5 Technical Lemmas
	6 The Case ν= 1
	7 The Case ν> 1
	8 The Analytic Case
	9 The Skew-Shift Case, ν> 1
	References
The Slicing Problem by Bourgain
	1 Introduction
	2 The Isotropic Position
	3 Distribution of Volume in Convex Bodies
	4 Bound for the Isotropic Constant
	References
On the Work of Jean Bourgain in Nonlinear Dispersive Equations
	1 Introduction
	2 Nonlinear Dispersive Equations: The Well-Posedness Theory Before Bourgain
	3 Bourgain's Transformative Work on the Well-Posedness Theory of Dispersive Equations
	4 A Quick Sampling of Some of the Other Groundbreaking Contributions of Bourgain to Nonlinear Dispersive Equations
		4.1 Gibbs Measure Associated to Periodic (NLS)
		4.2 Bourgain's ``High-Low Decomposition''
		4.3 Bourgain's Work on the Defocusing Energy Critical (NLS)
	5 Conclusion
	References
On Trace Sets of Restricted Continued Fraction Semigroups
	1 Introduction
		1.1 McMullen's Arithmetic Chaos Conjecture
		1.2 Thin Semigroups
		1.3 The Local-Global and Positive Density Conjectures
		1.4 Statements of the Main Theorems
		1.5 Notation
	2 Preliminary Remarks
	3 Proof of Theorem 1.5
	4 Proof of Theorem 1.6
	5 Proof of Lemma 1.9
	References
Polynomial Equations in Subgroups and Applications
	1 Introduction
		1.1 Background and Motivation
		1.2 New Results
	2 Solutions to Polynomial Equations in Subgroups of Finite Fields
		2.1 Stepanov's Method
		2.2 Some Divisibilities and Non-divisibilities
		2.3 Derivatives on Some Curves
		2.4 Multiplicity Points on Some Curves
	3 Small Divisors of Integers
		3.1 Smooth Numbers
		3.2 Number of Small Divisors of Integers
	4 Proof of Theorem 1.2
		4.1 Preliminary Estimates
		4.2 Optimization of Parameters
	5 Proof of Theorem 1.6
		5.1 Outline of the Proof
		5.2 Formal Argument
	6 Comments
	References
Exponential Sums, Twisted Multiplicativity, and Moments
	1 Introduction
		1.1 Exponential Sums with Polynomials
		1.2 Sums of Twisted Multiplicative Functions
		1.3 Non-correlation of Exponential Sums for Different Polynomials
		1.4 Previous Work
	2 Sums of Twisted Multiplicative Functions
	3 Exponential Sums of Polynomials: Preliminary Results
	4 Proof of Theorem 1.1
	5 The Fourth Moment: Proof of Theorem 1.3
	6 Generic Polynomials
	7 Multiple Correlations
	8 Remarks on Katz's Theorem
	References
The Ternary Goldbach Problem with a Missing Digit and Other Primes of Special Types
	1 Introduction
	2 Outline of the Proof
	3 Structure of the Paper
	4 Sieve Decomposition and Proof of Theorem 1.1
	5 Fourier Estimates and Large Sieve Inequalities
	6 Local Versions of Maynard's Results
	7 Sieve Asymptotics for Local Version of Maynard
	8 b-Variable Circle Method
	9 b-Variable Major Arcs
	10 Generic Minor Arcs
	11 Exceptional Minor Arcs
	12 The Ternary Goldbach Problem with a Prime with a Missing Digit, a Piatetski-Shapiro Prime, and a Prime of Another Special Type
	References
A Note on Harmonious Sets
	1 A Wrong Lemma Is Revisited
	2 Bogolyobov's Approach
	3 New Examples of Harmonious Sets
	4 The Union of Two Harmonious Sets
	References
On the Multiplicative Group Generated by Two Primes in Z/QZ
	1 Introduction
		1.1 Notation
	2 Proof of Theorem 4
	References