Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.26146

Prescribed Beurling densities in spectral eigen-subspaces

Zheng-Yi Lu
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $(q,D,L)$ be a one-dimensional Hadamard triple with $2\leq m:=\#D<q$, and let $μ_{q,D}$ be the associated self-similar probability measure. Write $\mathcal T=\{t\in\mathbb Z:(q,D,tL)\text{ is a Hadamard triple}\}$. We prove that, for every $0<s<\log m/\log q$ and every $ζ\in[0,+\infty]$, there are continuum many spectra $Λ$ of $μ_{q,D}$ such that $tΛ$ is a spectrum for every $t\in\mathcal T$, $\dim_{\mathrm{Be}}Λ=s$, and $D_s^+(Λ)=ζ$. The construction combines a nested family of finite digit sets with lacunary spectral extensions. A two-sided interval-counting estimate controls the density of the prescribed subset, while the added frequencies ensure completeness for every multiplier without changing that density. Binary choices in the extension yield continuum many distinct spectra. This refines Beurling dimension interpolation in spectral eigen-subspaces by prescribing the upper density at every strictly intermediate dimension.

Explore related subjects

math.FAmath.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Zheng-Yi Lu. 2026-10-19. Prescribed Beurling densities in spectral eigen-subspaces. https://arxiv.org/abs/2610.26146

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

KEEP EXPLORING

Related papers

arXiv2026

Linear Preservers of Semipositivity over Proper Cones

We classify all real linear operators on $M_{m,n}(\mathbb{R})$, with $m,n\ge2$, that preserve semipositivity between two prescribed pairs of proper cones, without assuming invertibility. Every such operator has product form $L(A)=XAY$ or degenerate form $L(A)=Ψ(u^TA)$, possibly both. We give necessary and sufficient cone conditions for each form; the degenerate condition is expressed through a linear matrix pencil on the dual of the target output cone. The degenerate preservers are exactly those of operator rank $n$, and every preserver of larger rank has product form. We also classify all linear preservers from $M_{2,2}(\mathbb{R})$ to $M_{1,3}(\mathbb{R})$ over the nonnegative orthants, obtaining three disjoint classes, including two rank-three classes outside product form.

math.FA↗
arXiv2026

Geometric conditions which imply $w^*$-normal structure

Let $X$ be a complex Banach space. In their seminal work, Lau and Mah (\cite{LM}) study a geometric property called Lim's condition which, for separable $X$, implies that for any non-empty $w^*$-compact convex set $K \subset X^*$, any nonexpansive mapping $T: K \rightarrow K$ has a fixed point. In this paper, we study a weaker form of Lim's condition, which we call weak-Lim's condition. We prove that if $\ell_1 \not\hookrightarrow X$, then weak-Lim's condition on $X^*$ implies the fixed point property for non-expansive maps on weakly compact subsets of $X$. In the case of a commutative $C^\ast$-algebra, we prove that weak-Lim's condition is equivalent to Lim's condition. We show that Lim's condition characterizes $C^*$-algebras which are a $c_0$-direct sum of finite-dimensional operator spaces, while weak-Lim's condition characterizes $C^*$-algebras which are a $c_0$-direct sum of the spaces of compact operators on Hilbert spaces. For a uniform algebra, we extend a result of Lau and Mah to show that weak-Lim's condition implies that the space is finite-dimensional. In the case of an $L^1$-predual space, we show that this condition implies $k$-smoothness of the norm in the sense considered in \cite{LR}.

math.FA↗
arXiv2026

Kubo-Ando Means of Finite Order over Real Division Algebras

In this paper, we study Kubo-Ando means of finite order over the real division algebras $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$. We establish an explicit bijection between Kubo-Ando means of order $n$ and normalized positive functions $f$ such that both $f$ and its transpose $f^{\top}(t)=tf(t^{-1})$ are matrix monotone of order $n$. We also prove that matrix monotonicity of a fixed order is independent of the underlying division algebra. We then investigate finite-order Kubo-Ando means under the canonical embeddings relating quaternionic, complex, and real positive definite matrices. These embeddings preserve functional calculus and the Loewner order, inducing natural correspondences between means on the original and embedded cones. We characterize their extensions to the ambient cones and study the geometry of the embeddings with respect to the Log-Euclidean metric. As an application, we obtain explicit affine formulas for Kubo-Ando means of order $2$ on $\mathscr{P}_{2}(\mathbb{D})$ and transfer them to the associated embedded cones of $4\times4$ positive definite matrices. In particular, we derive trace-determinant formulas for the geometric mean and show that they do not, in general, extend to the full ambient cones.

math.FA↗