arXiv subjects

Jinjun Li

Publications and source records attributed to Jinjun Li.

10 recordsLinked to original sources

arXiv2026

Macroscopic return-time spectra and Hausdorff dimension transitions

Can the complete macroscopic spectrum of return times determine the Hausdorff dimension of the corresponding set of initial points? We classify all common lower and upper macroscopic intermediate-dimension spectra of zero-density integer sets and realize every admissible spectrum on a full-dimensional family of zero-frequency orbits under logarithmic-power bounds on inverse-branch contraction. For first-return branches with stretched-exponential contraction $e^{-n^δ}$, $0<δ<1$, complete spectrum levels exhibit a sharp zero/full dimension transition at the counting endpoint $δ$. At the critical exponent, exact laws $N_y(t)\sim c_0t^δ(\log t)^β$ yield zero dimension for $β\le0$ and full dimension for $β>0$, even with the complete spectrum fixed. An explicit two-branch intermittent interval map realizes this transition through genuine first returns. Above the critical exponent, imposing an exact counting law and a complete spectrum does not reduce the dimension of an intrinsic mean-energy level in an affine return tower. For nonlinear branches, we obtain quantitative bounds and an exact conditional-pressure formula under thermodynamic regularity assumptions, verified for an explicit non-affine family.

math.CA↗

arXiv2026

Discrete intermediate dimensions: endpoints, spectra and quasi-isometric invariance

We introduce upper and lower discrete intermediate dimensions for unbounded subsets of Euclidean space and extend them to metric spaces. The definitions restrict the covering radii within a scale-dependent range of dyadic shells and take a supremum over an auxiliary parameter. The resulting dimensions are monotone in the parameter and recover the Barlow--Taylor discrete Hausdorff dimension at zero. We prove continuity at every positive parameter, with a sharp quantitative bound, and give an example with a jump at zero. At the upper endpoint, the upper dimension equals upper mass dimension for uniformly discrete Euclidean sets; the analogous lower mass identity fails. We establish a mass distribution principle and prove quasi-isometric invariance under a uniform coarse packing condition. For spaces with uniform polynomial volume growth, we construct clustered subsets attaining the continuity bound. Independent thinning truncates these spectra at the resulting mass dimension, almost surely for all parameters simultaneously. Applications include the discrete Heisenberg group with its word metric and the ranges of simple symmetric random walks.

math.CA↗

arXiv2026

Fourier Restriction Beyond Dimension Data: A Multiscale Moment Criterion

We establish an exact Fourier restriction criterion for a class of weighted Moran measures. Assuming uniformly bounded one-level frame bounds, we prove that, for every $s>2$, the measure itself already detects the full extension estimate: $L^2(μ)\to L^s(\mathbb R)$ extension holds if and only if $\widehatμ\in L^s(\mathbb R)$, equivalently, if and only if the one-level Fourier moments $G_{s,n}$ are summable, where $G_{s,n}$ is the $s$th moment of the nonzero Fourier coefficients of the $n$th digit law. The sufficient direction is obtained from a finite-frame stability estimate and its multiscale iteration, while necessity follows from disjoint resonant frequency grids. This gives, in particular, an exact formula for the critical restriction exponent and a precise criterion for endpoint attainment. The criterion reveals that the optimal restriction range is not determined by standard Fourier or dimensional data. We construct singular continuous spectral Moran measures with the same scales, digit cardinalities, exponential spectrum, and maximal nonzero Fourier coefficients. They also satisfy the same global Fourier decay bound, which is sharp along the same sequence of resonant frequencies, yet their critical restriction exponents are different. More generally, after fixing the summability rate of the maximal Fourier coefficients, we determine the optimal interval of possible restriction exponents and show that, at every interior critical exponent, the same fixed data allow both attainment and failure of the critical endpoint. These measures may simultaneously have $ \dim_{\mathrm F}^θμ=θ$ and $ D_q(μ)=1,$for all admissible $θ$ and $q$, and pointwise local dimension one everywhere on their supports.

math.CA↗

arXiv2026

Projection theorems for macroscopic intermediate dimensions

We introduce lower and upper macroscopic intermediate dimensions of unbounded subsets of Euclidean space by combining covering contributions from all dyadic annuli beyond a given scale. For every unbounded Borel set $E\subset\R^d$ and $m\in \{1,\ldots,d-1\}$, we prove that the macroscopic intermediate dimensions of $P_VE$ equal the corresponding intrinsic capacity profiles for almost every $m$-dimensional linear subspaces $V\subset \R^d$. One full-measure set works for both dimensions and every parameter $θ\in[0,1]$. The proof combines a covering--capacity comparison with an exponent gap, fractional-moment estimates for projected coverings, and capacity-weighted retention of source shells. The dimensions recover macroscopic Hausdorff dimension at parameter zero, are invariant under quasi-isometries, and are continuous at positive parameters. The upper macroscopic intermediate dimension agrees with the upper discrete intermediate dimension introduced in \cite{LXZ}, whereas the lower one can differ. We compute the projected dimensions of bounded-base discrete digit Moran sets, including the self-similar case obtained when the bases and digit sets are constant, and give applications to Lipschitz graphs and arithmetic sumsets.

math.CA↗

arXiv2026

Spectrality of factors of product spectral measures

We refine the method by Greenfeld and Lev for the product spectral set problem and generalize the theorem to a singular measure setting. Furthermore, we establish a new class of spectral unions of intervals for which the product spectral set question has a positive answer. More precisely, if $A$ is a subset of the natural numbers such that $A\oplus B = \{0,1,\cdots, N-1\}$ for some $B\subset \mathbb N$ and $N>1$ then the product measure $\mathcal{L}|_{A+[0,1]}\times ν$ is a spectral measure (that may be singular) if and only if $ν$ is a spectral measure.

math.CA↗

arXiv2026

Quantitative Hardy--Littlewood maximal inequalities and Wiener--Stein theorem on p.c.f. fractals

Let $K\subset \mathbb{R}^d$ be a post-critically finite (p.c.f.) self-similar set with Hausdorff dimension $s$, and $μ$ be a self-similar probability measure supported on $K$. Let $H^α_μ$, $0<α\le s$, be the Hausdorff content on $K$, and $M_{\mathcal{D}}^μ$ be the Hardy--Littlewood maximal operator defined on $K$ associated with its basic cubes $\mathcal{D}$. In this paper, we establish quantitative strong type and weak type Hardy--Littlewood maximal inequalities on fractal set $K$ with respect to $H^α_μ$ for all range $0<α\le s$. As applications, the Lebesgue differentiation theorem on $K$ is proved. Moreover, via the Hardy--Littlewood maximal operator $M_{\mathcal{D}}^μ$, we characterize the Lebesgue--Choquet space $L^p(K,H^α_μ)$ and the Zygmund space $L\log L(K,μ)$. To be exact, given $α/s< p\le \infty$, we discover that \[ \text{$f\in L^p(K,H^α_μ)$ if and only if $M_{\mathcal{D}}^μf\in L^p(K,H^α_μ)$}\] and, for $f\in L^1(K,μ)$ with $K$ satisfying the strong separation condition, \[\text{$M_{\mathcal{D}}^μf\in L^1(K,μ)$ if and only if $f\in L\log L(K,μ)$}.\] That is, Wiener's $L\log L$ inequality and its converse inequality due to Stein in 1969 are extended to fractal set $K$ with respect to $μ$.

math.FA↗

arXiv2023

Spectra of the Sierpiński type spectral measure and their Beurling dimensions

In this paper, we study the structure of the spectra for the Sierpiński type spectral measure $μ_{A,\mathcal{D}}$ on $\mathbb{R}^2$. We give a sufficient and necessary condition for the family of exponential functions $\{e^{-2πi\langleλ, x\rangle}: λ\inΛ\}$ to be a maximal orthogonal set in $L^2(μ_{A,\mathcal{D}})$. Based on this result, we obtain a class of regular spectra of $μ_{A,\mathcal{D}}$. Moreover, we discuss the Beurling dimensions of the spectra and obtain the optimal upper bound of Beurling dimensions of all spectra, which is in stark contrast with the case of self-similar spectral measure. An intermediate property about the Beurling dimension of the spectra is obtained.

math.NT↗

arXiv2023

On the intermediate value property of spectra for a class of Moran spectral measures

We prove that the Beurling dimensions of the spectra for a class of Moran spectral measures are between $0$ and their upper entropy dimensions. Moreover, for such a Moran spectral measure $μ$, we show that the Beurling dimension for the spectra of $μ$ has the intermediate value property: let $t$ be any value between $0$ and the upper entropy dimension of $μ$, then there exists a spectrum whose Beurling dimension is $t.$ In particular, this result settles affirmatively a conjecture involving spectral Bernoulli convolution proposed by Fu, He and Wen in [J. Math. Pures Appl. 116 (2018), 105--131]. Furthermore, we prove that the set of the spectra whose Beurling dimensions are equal to any fixed value between $0$ and $\ue μ$ has the cardinality of the continuum.

math.NT↗

arXiv2018

On Assouad dimension and arithmetic progressions in sets defined by digit restrictions

We show that the set defined by digit restrictions contains arbitrarily long arithmetic progressions if and only if its Assouad dimension is one. Moreover, we show that for any $0\le s\le 1$, there exists some set on $\mathbb{R}$ with Hausdorff dimension $s$ whose Fourier dimension is zero and it contains arbitrarily long arithmetic progressions.

math.CA↗

arXiv2016

On exceptional sets in Erdős-Rényi limit theorem revisited

For $x\in [0,1],$ the run-length function $r_n(x)$ is defined as the length of the longest run of $1$'s amongst the first $n$ dyadic digits in the dyadic expansion of $x.$ Erdős and Rényi proved that $\lim\limits_{n\to\infty}\frac{r_n(x)}{\log_2n}=1$ for Lebesgue almost all $x\in[0,1]$. Let $H$ denote the set of monotonically increasing functions $φ:\mathbb{N}\to (0,+\infty)$ with $\lim\limits_{n\to\infty}φ(n)=+\infty$. For any $φ\in H$, we prove that the set \[ E_{\max}^φ=\left\{x\in [0,1]:\liminf\limits_{n\to\infty}\frac{r_n(x)}{φ(n)}=0, \limsup\limits_{n\to\infty}\frac{r_n(x)}{φ(n)}=+\infty\right\} \] either has Hausdorff dimension one and is residual in $[0,1]$ or empty. The result solves a conjecture posed in \cite{LW5} affirmatively.

math.PR↗