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Christoph Thiele

Publications and source records attributed to Christoph Thiele.

At least 19 recordsLinked to original sources
arXiv2026

A blueprint for the formalization of norm-variation of multiple ergodic averages for commuting transformations

This blueprint serves as a companion to a forthcoming, shorter traditional mathematical paper. The purpose of this blueprint is two-fold: first, it has served as the foundation for a formalization in Lean 4 of these results. This formalization has been completed largely automatically, making essential use of current frontier large language models. Second, it will serve as a resource to readers of the main paper who are interested in further technical details of the proofs. The main result concerns norm-variation estimates for multiple ergodic averages associated with $n\ge 2$ commuting measure preserving transformations, providing a quantitative strengthening of Tao's norm-convergence theorem and answering an open question of Avigad and Rute. At the core of the analysis lies an explicit real-variable estimate for twisted multilinear averages that is closely related to certain singular Brascamp--Lieb inequalities.

math.DS↗
arXiv2026

Quantitative Polynomial Wiener-Wintner Theorems

We prove quantitative polynomial Wiener--Wintner theorems in a very general setup, including measure-preserving actions of nilpotent Lie groups. Our results apply both to ergodic averages and to averages with singular integral weights. The proof relies on the generalized polynomial Carleson theorem developed in the companion paper by van Doorn, Srivastava, and the authors.

math.DS↗
arXiv2026

Balancing label resolution and computational cost in dynamical models of lipid metabolism

Lipid metabolism is a central biological process that is commonly studied using destructive mass-spectrometry experiments. A recently proposed strategy, uses multiple labels to extract temporal information about lipid metabolism from a single destructive measurement. However, the computational complexity of the model-based data analysis increases rapidly with the number of labels, creating a fundamental trade-off between the information content of the measurements and the cost of analysis. Here, we examine how the number of modelled labels affects parameter estimation accuracy, trajectory recovery, and computational cost, and whether modelling fewer labels than are experimentally available can mitigate this trade-off. Using synthetic data from a five-label experiment, we find that modelling three of the five labels provides a practical balance between experimental feasibility, inferential power, and computational tractability. In an application to hepatocyte triglyceride cycling, we further show that the most cost-efficient, single-label model can yield biologically implausible predictions for unobserved species, whereas models that resolve more labels better constrain these latent dynamics. These results provide practical guidance for selecting model resolution in multi-label experiments and establish a quantitative basis for balancing inferential power against computational cost.

q-bio.QM↗
arXiv2026

An SU(2n)-valued nonlinear Fourier transform

We define a nonlinear Fourier transform which maps sequences of contractive $n \times n$ matrices to $SU(2n)$-valued functions on the circle $\mathbb{T}$. We characterize the image of finitely supported sequences and square-summable sequences on the half-line, and construct an inverse for $SU(2n)$-valued functions whose diagonal $n \times n$ blocks are outer matrix functions. As an application, we relate this nonlinear Fourier transform with quantum signal processing over $U(2n)$ and multivariate quantum signal processing.

math.CA↗
arXiv2026

Infinite quantum signal processing for arbitrary Szegő functions

We provide a complete solution to the problem of infinite quantum signal processing for the class of Szegő functions, which are functions that satisfy a logarithmic integrability condition and include almost any function that allows for a quantum signal processing representation. We do so by introducing a new algorithm called the Riemann-Hilbert-Weiss algorithm, which can compute any individual phase factor independent of all other phase factors. Our algorithm is also the first provably stable numerical algorithm for computing phase factors of any arbitrary Szegő function. The proof of stability involves solving a Riemann-Hilbert factorization problem in nonlinear Fourier analysis using elements of spectral theory.

quant-ph↗
arXiv2026

Uniform bounds for bilinear symbols with linear K-quasiconformally embedded singularity

We prove bounds in the strict local $L^{2}(\mathbb{R}^{d})$ range for trilinear Fourier multiplier forms with a $d$-dimensional singular subspace. Given a fixed parameter $K \ge 1$, we treat multipliers with non-degenerate singularity that are push-forwards by $K$-quasiconformal matrices of suitable symbols. As particular applications, our result recovers the uniform bounds for the one-dimensional bilinear Hilbert transforms in the strict local $L^{2}$ range, and it implies the uniform bounds for two-dimensional bilinear Beurling transforms, which are new, in the same range.

math.CA↗
arXiv2025

Carleson operators on doubling metric measure spaces

Doubling metric measure spaces provide a natural framework for singular integral operators. In contrast, the study of maximally modulated singular integral operators, the so-called Carleson operators, has largely been limited to Euclidean space with modulation functions such as polynomials defined by algebraic means. We present a general axiomatic approach to modulation functions on doubling metric measure spaces and prove $L^p$ bounds for the corresponding Carleson operators in Theorem 1.1 and Theorem 1.2. This generalizes classical and modern results on Carleson operators. In addition to the proofs presented here, our main results have been computer verified using the language Lean and the library mathlib, as documented in the sibling communication arXiv:2405.06423.

math.CA↗
arXiv2025

A blueprint for the formalization of Carleson's theorem on convergence of Fourier series

This paper is the blueprint underlying the Lean formalization of the proof of Carleson's classical result asserting almost everywhere convergence of Fourier series of continuous functions. We break up the proof into two steps, a reduction of the classical result to a new theorem that appears in a sibling communication and a proof of this new theorem, which is also detailed as blueprint in this paper. An early version of this blueprint was used to initiate the Lean formalization. During the formalization, many contributors elaborated the blueprint with minor corrections, modifications and extensions. The final version is presented here as a guide through the accompanying Lean code.

math.CA↗
arXiv2025

One sided orthogonal polynomials and a pointwise convergence result for $SU(2)$-valued nonlinear Fourier series

We elaborate on a connection between the $SU(2)$-valued nonlinear Fourier series and sequences of left and right orthogonal polynomials for complex measures on the unit circle. We show a convergence result for the associated reproducing kernel. This is a universality type result in the vein of Mate-Nevai-Totik, which turns out to be much simpler in the $SU(2)$ case than in the $SU(1,1)$ case. We then relate a.e. pointwise convergence of the product of left and right polynomials and their squares with both behavior of their zeros as well as behavior of some local parameters for these polynomials. We conclude by proving almost everywhere convergence along lacunary sequences of the functional $(a_n ^* +b_n)(a_n - b_n ^*)$ of the partial $SU(2)$-valued nonlinear Fourier series $(a_n, b_n)$ under the assumption that the nonlinear Fourier series $(a,b)$ itself satisfies both $\|b\|_{L^{\infty} (\mathbb{T})} < 2^{- \frac 1 2}$ and $a^*$ is outer.

math.CA↗
arXiv2025

Norm-variation of triple ergodic averages for commuting transformations

We prove an $r$-variation estimate, $r>4$, in the norm for ergodic averages with respect to three commuting transformations. It is not known whether such estimates hold for all $r\ge 2$ as in the analogous cases for one or two commuting transformations, or whether such estimates hold for any $r<\infty$ for more than three commuting transformations.

math.CA↗
arXiv2024

Maximal multilinear operators

We establish multilinear $L^p$ bounds for a class of maximal multilinear averages of functions on one variable, reproving and generalizing the bilinear maximal function bounds of Lacey. As an application we obtain almost everywhere convergence results for these averages, and in some cases we also obtain almost everywhere convergence for their ergodic counterparts on a dynamical system.

math.CA↗
arXiv2024

Quantum signal processing and nonlinear Fourier analysis

Elucidating a connection with nonlinear Fourier analysis, we extend a well known algorithm in quantum signal processing to represent measurable signals by square summable sequences. Each coefficient of the sequence is Lipschitz continuous as a function of the signal.

quant-ph↗
arXiv2024

Phase space localizing operators

We construct phase space localizing operators in all dimensions. These are frequency localized variants of the conditional expectation operator related to a dyadic stopping time. Our construction is an improvement over the so-called phase plane projections of Muscalu, Tao and the third author in one dimension. The motivation for such operators comes from time-frequency analysis. They are used in particular to prove uniform estimates for multilinear modulation invariant operators.

math.CA↗
arXiv2022

When does $e^{-\lvert τ\rvert }$ maximize Fourier extension for a conic section?

In the past decade, much effort has gone into understanding maximizers for Fourier restriction and extension inequalities. Nearly all of the cases in which maximizers for inequalities involving the restriction or extension operator have been successfully identified can be seen as partial answers to the question in the title. In this survey, we focus on recent developments in sharp restriction theory relevant to this question. We present results in the algebraic case for spherical and hyperbolic extension inequalities. We also discuss the use of the Penrose transform leading to some negative answers in the case of the cone.

math.CA↗
arXiv2022

Local bounds for singular Brascamp-Lieb forms with cubical structure

We prove a range of $L^p$ bounds for singular Brascamp-Lieb forms with cubical structure. We pass through sparse and local bounds, the latter proved by an iteration of Fourier expansion, telescoping, and the Cauchy-Schwarz inequality. We allow $2^{m-1}<p\le \infty$ with $m$ the dimension of the cube, extending an earlier result that required $p=2^m$. The threshold $2^{m-1}$ is sharp in our theorems.

math.CA↗