Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.38909

Uncertainty principles as forcing mechanisms

Laurent Massoulié
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this note we first consider the Heisenberg uncertainty principle. We show that it acts as a {\bf forcing mechanism} in the following sense. Real functions that approach its limit are close to Gaussian distributions, with proximity both in the sense of total variation distance and Wasserstein-1 distance. We next consider the uncertainty principle for functions defined on finite Abelian groups, and show that it acts as a forcing mechanism in the following sense. Functions that approach its limit are close (after translation and modulation by a complex exponential) to indicator functions of some subgroup, with proximity measured in ${\mathbb {L}}^1$-norm.

Explore related subjects

math.CAmath.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Laurent Massoulié. 2026-10-28. Uncertainty principles as forcing mechanisms. https://arxiv.org/abs/2610.38909

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Hausdorff dimension of the graph of a random vector-valued Weierstrass function

Let $\{θ_n,λ_n:n\geq0\}$ be mutually independent random variables, each uniformly distributed on $[0,1]$, and write $Θ=(θ_n)_{n\geq0}$ and $Λ=(λ_n)_{n\geq0}$. The random vector-valued Weierstrass function is given by $$f_{Θ,Λ}(x)= \left( \sum_{n=0}^{\infty} a^n\cos\bigl(2π(b^n x+θ_n)\bigr),\ \sum_{n=0}^{\infty} a^n\sin\bigl(2π(b^n x+λ_n)\bigr) \right), \; x\in[0,1],$$ where $0 1$. The Hausdorff dimension of the graph of this function is proved to be $$\dim_H G(f_{Θ,Λ}) = \min\left\{-\frac{\log b}{\log a}, \, 3 +2\frac{\log a}{\log b}\right\} \quad \text{a.s}.$$

math.CA↗
arXiv2026

Characterization of the boundedness of the segment multiplier on rearrangement-invariant spaces

Given rearrangement-invariant (r.i.) spaces $X$ and $Y$, we show that the segment multiplier, the truncated Hilbert transform, and the discrete Hilbert transform (on the associated discretized space) are bounded from $X$ to $Y$ simultaneously. Moreover, when $X = Y$ this boundedness is characterized by a condition on a pair of Boyd-type indices. We also exhibit an r.i. space on which the segment multiplier is bounded while the (non-truncated) Hilbert transform fails to be bounded.

math.CA↗
arXiv2026

Endpoint regularity of Radon transforms associated with real analytic phases in the plane

In this paper, we consider the endpoint Sobolev regularity of Radon transforms associated with real-analytic functions in the plane. We establish all of the failure of endpoint regularity for real-analytic phases. On the one hand, the critical endpoint regularity $L^p(\mathbb{R}^2)\rightarrow L^p_s(\mathbb{R}^2)$, corresponding to extreme points of the reduced Newton polyhedron of phase function, is true if only and only if $p=2$. For generic extreme points, this failure of endpoint regularity generalizes a result of Christ \cite{christ}. On the other hand, we find that the endpoint Sobolev regularity also fails on a bounded segment of the reduced Newton polyhedron when the bisectrix $y=x$ intersects one of unbounded faces at a relative interior point. In particular, our results imply that the Radon transform is not bounded from $L^p$ into $L^p_{1/n}$ with $p=n/(n-1)$ or $p=n$, where the phase functions are real-valued homogeneous polynomials $S(x,y)=a_1x^{n-1}y+a_2x^{n-2}y^2+\cdots+a_{n-1}xy^{n-1}$ with $n\geq 3$ and $a_1a_{n-1}\neq0$. This answers a problem of Phong and Stein in 1994. Moreover, the main results of this paper also give examples which show failure of critical Sobolev regularity for degenerate Fourier integral operators satisfying finite type conditions formulated in \cite{Seeger93,seeger2}. Based on earlier results and our proof of several remaining cases, we conclude that all of other critical endpoint Sobolev estimates are true for Radon transforms associated with real-analytic phases in the plane. Therefore, all failures of endpoint regularity estimates are clearly determined.

math.CA↗