arXiv subjects

Guozheng Cheng

Publications and source records attributed to Guozheng Cheng.

8 recordsLinked to original sources

arXiv2026

A quantitative form of the planar Falconer distance theorem

We give a quantitative planar form of a multiscale argument for Falconer's distance problem. For every $0<η<1$, we obtain a positive lower bound for the Lebesgue measure of the distance set of a compact subset of the unit ball supporting a probability measure with $ν(B(x,r))\le 2r^{1+η}$. The bound depends only on $η$. For a compact set of diameter $D$ carrying a probability measure with $μ(B(x,r))\le A(r/D)^{1+\varepsilon}$, localization gives $|ΔE|\ge c_{\varepsilon,p}DA^{-p}$ for every fixed $p>1/\varepsilon$. In particular, compact planar sets of Hausdorff dimension greater than one have distance sets of positive Lebesgue measure. The proof develops the multiscale method of the September 2026 manuscript of OpenAI, using one sequence of restricted distance measures for both the mass and Fourier estimates. A reconstruction lemma combines summable total-variation errors with summable squared Fourier norms. No endpoint or sharpness assertion is made.

math.CA↗

arXiv2026

Strict Hausdorff-dimension drop for Quartic Salem Bernoulli convolutions

For every quartic Salem number $β$, we prove that the equal-weight Bernoulli convolution $ν_{β^{-1}}$ has Hausdorff dimension strictly less than one. The main step converts large Fourier coefficients at geometrically separated frequencies into a deficit in the Shannon entropy of a finite convolution. A positive trigonometric product gives the entropy estimate, while the algebraic norm controls the number of distinct atoms in each spatial cell. For the two quartic Salem numbers in $(1,2)$, reciprocal differences at suitable return times produce the required Fourier coefficients. A finite block cover then gives the dimension bound. This proof is independent of numerical computation. Separate exact calculations give $\dimHν_{β_1^{-1}}<1-1.4\cdot10^{-23}$ and $\dimHν_{β_2^{-1}}<1-7.8\cdot10^{-22}$, where $β_1<β_2$ are the two parameters in $(1,2)$. The accompanying verification programs use rational interval arithmetic and the Python standard library.

math.DS↗

arXiv2026

Strict correlation-dimension drop for quartic Salem Bernoulli convolutions

For every quartic Salem number $β\in(1,2)$, we prove that the Bernoulli convolution $ν_{β^{-1}}$ with equal weights has correlation dimension strictly less than one. In particular, neither of the two corresponding measures has an $L^2$ density. We relate correlation dimension to exact collision probabilities and express these probabilities as positive integrals over a four-dimensional torus associated with the defining polynomial. The unit-circle conjugates produce a persistent oscillatory term; using carefully chosen return times of the corresponding rotation, we take differences that make this term small while controlling the length and cost of the resulting configurations. Many well-separated placements of these configurations then give a collision probability above the critical scale, which yields the strict dimension drop. The qualitative argument is independent of the computer-assisted estimates. Separate computer-assisted arguments give \[ 0.99999<D_2(ν_{β_1^{-1}})<1-10^{-65}, \qquad 0.99999<D_2(ν_{β_2^{-1}})<1-10^{-75}, \] where $β_1<β_2$ are the two quartic Salem numbers in $(1,2)$.

math.DS↗

arXiv2026

Canonical Mandelbrot Cascades on Curves Are Rajchman

We settle the Rajchman problem for canonical scalar dyadic Mandelbrot cascades at the minimal Kahane--Peyrière integrability threshold. If $μ$ is the cascade on $[0,1]$, then $\widehatμ(ξ)\to 0$ as $|ξ|\to\infty$, almost surely on non-extinction. For every fixed nondegenerate $C^2$ embedded arc $γ:[0,1]\to\mathbb{R}^2$, the pushforward $γ_\#μ$ is likewise Rajchman almost surely on non-extinction. The analogous conclusion holds for the scalar cascade on the parameter circle pushed forward by any fixed nondegenerate $C^2$ Jordan curve. No moment condition of order strictly greater than one is imposed; in particular, the results include the regime $\mathbb{E}[W^q]=\infty$ for every $q>1$. The proof combines a spine-based lower-deviation principle, adaptive terminal approximation, and predictable capping to obtain almost-sure estimates uniform over large frequency annuli without higher moments. For curved pushforwards, an endpoint-safe phase decomposition controls direction-dependent stationary regions, including those meeting the endpoints of an arc, and couples the geometric and probabilistic arguments through a common dyadic kernel. Combined with the exact Fourier-dimension formulas for the corresponding models, the theorems show that Rajchman decay persists at zero Fourier dimension.

math.PR↗

arXiv2026

Exact Fourier dimensions of dyadic Mandelbrot cascades under minimal integrability

We determine the Fourier dimension of dyadic Mandelbrot cascades under the minimal Kahane-Peyriere integrability condition. The interval theorem is proved in a vector-valued dyadic cascade model in which sibling weights may have arbitrary dependence. For every balanced energy-admissible vector law, almost surely on non-extinction, dim_F(mu)=dim_E(mu)=dim_2(mu)=D_E(X). In the canonical scalar case, under W>=0, E W=1, E[W log_2^+ W] 1} max{0, (q-1-log_2 E[W^q])/q}. The interval and circle formulas share a light-tail/heavy-tail dichotomy but have different mechanisms: energy dimension for the interval, and minimum lower local dimension for the circle. The circle lower bound follows from a finite-moment annular Fourier theorem.

math.PR↗

arXiv2021

A Gaussian version of Littlewood's theorem on random power series

We prove a Littlewood-type theorem on random analytic functions for not necessarily independent Gaussian processes. We show that if we randomize a function in the Hardy space $H^2(\dd)$ by a Gaussian process whose covariance matrix $K$ induces a bounded operator on $l^2$, then the resulting random function is almost surely in $H^p(\dd)$ for any $p>0$. The case $K=\text{Id}$, the identity operator, recovers Littlewood's theorem. A new ingredient in our proof is to recast the membership problem as the boundedness of an operator. This reformulation enables us to use tools in functional analysis and is applicable to other situations. The sharpness of the new condition and several ramifications are discussed.

math.FA↗

arXiv2018

Random weighted shifts

In this paper we initiate the study of a fundamental yet untapped random model of non-selfadjoint, bounded linear operators acting on a separable complex Hilbert space. We replace the weights $w_n=1$ in the classical unilateral shift $T$, defined as $Te_n=w_ne_{n+1}$, where $\{e_n\}_{n=1}^\infty$ form an orthonormal basis of a complex Hilbert space, by a sequence of i.i.d. random variables $\{X_n\}_{n=1}^{\infty}$; that is, $w_n=X_n$. This paper answers basic questions concerning such a model. We propose that this model can be studied in comparison with the classical Hardy/Bergman/Dirichlet spaces in function-theoretic operator theory. We calculate the spectra and determine their fine structures (Section 3). We classify the samples up to four equivalence relationships (Section 4). We introduce a family of random Hardy spaces and determine the growth rate of the coefficients of analytic functions in these spaces (Section 5). We compare them with three types of classical operators (Section 6); this is achieved in the form of generalized von Neumann inequalities. The invariant subspaces are shown to admit arbitrarily large indices and their semi-invariant subspaces model arbitrary contractions almost surely. We discuss a Beurling-type theorem (Section 7). We determine various non-selfadjoint algebras generated by $T$ (Section 8). Their dynamical properties are clarified (Section 9). Their iterated Aluthge transforms are shown to converge (Section 10). In summary, they provide a new random model from the viewpoint of probability theory, and they provide a new class of analytic functional Hilbert spaces from the viewpoint of operator theory. The technical novelty in this paper is that the methodology used draws from three (largely separate) sources: probability theory, functional Hilbert spaces, and the approximation theory of bounded operators.

math.FA↗

arXiv2018

Three Remarks on Carleson Measures for Dirichlet Space

In this paper, we prove that all doubling measures on the unit disk $\mathbb{D}$ are Carleson measures for the standard Dirichlet space $\mathcal{D}$. The proof has three ingredients. The first one is a characterization of Carleson measures which holds true for general reproducing kernel Hilbert spaces. The second one is another new equivalent condition for Carleson measures, which holds true only for the standard Dirichlet space. The third one is an application of the dyadic method to our settings.

math.FA↗