Variance of Lattice Point Counting in Thin Annuli

We determine the sharp weighted power-mean comparisons for the family $\{H_a(r)\}_{r\in (0,1)}$ with the logarithmic interpretation at $a=0$. In the parameter ranges considered by Barnard, Richards and Tiedeman, we give a complete characterization of all orders $λ,μ\in\mathbb{R}$ for which $$A_λ(w;1,1-r)\leq H_a(r)\leq A_μ(w;1,1-r)$$ holds for every $r\in(0,1)$. In particular, our results settle completely their two power-mean conjectures. The two sharp orders are determined by the second-order expansion at $r=0$ and the endpoint matching as $r\to1$. A weighted Wronskian identity and a sign analysis of its residual establish the global inequalities.

math.CA↗

arXiv2026

Spectral eigenvalue problem of Cantor measures and Artin's primitive root conjecture

The eigenvalue problem for a probability measure $μ$ with compact support in $\R$ is whether there exist a countable set $Λ$ and a nonzero real $t\ne 1$ such that both $Λ$ and $tΛ$ are spectra of $μ$, that is, the family $$E_{aΛ}=\{e^{-2πi aλx}:λ\inΛ\}$$ is an orthonormal base for $L^2(μ)$ for $a=1, t$. The eigenvalue problem was discovered independently by Strichartz \cite{Str00}, Łaba and Wang \cite{LW02} for the Cantor measures $μ_{4,\{0,1\}}$ and $μ_{6,\{0,1,2\}}$, respectively. In this paper, we investigate the spectral eigenvalue problem for the general spectral Cantor measure $μ_{b,\mathcal{D}}$. This topic is naturally related to elementary number theory. Unexpectedly, however, our main results depend on the theory of integers, especially Artin's primitive root conjecture. To some extent, our results suggest that Artin's primitive root conjecture may hold and confirms some viewpoints implied by Minkowski in \cite{Min57}.

math.CA↗

arXiv2026

Positive Bidiagonal Factorizations: Combinatorial Structure, Spectral Theory, and Matrix Continued Fractions

Prescribed positive bidiagonal factorizations of semi-infinite banded matrices reveal spectral, approximation, and integrable structures for arbitrary lower and upper bandwidths. Every cyclic permutation of the factors admits a normalized lower--upper positive factorization. Under suitable degree conditions, factor transfers induce matrix Christoffel transformations, linking the cyclic Darboux orbit to mixed-type Favard theory. Christoffel words determine minimum-height positive refactorizations and the cyclic products that reach them. The associated matrices have two block displacements, reducing to two diagonals for coprime bandwidths. Their finite nonzero spectra lie on a star, and planar networks yield positive radial Stieltjes moment sequences. In the compact radial case star-supported representing measures are characterized by conditions at the origin and negative fractional moments. For coprime bandwidths, a radial moment deformation gives determinant solutions of a sparse Lax hierarchy and synchronizes the cyclic Toda flows. Retaining the prescribed factor order yields matrix continued fractions with explicit Padé-type contact, denominator factorizations, backward evaluation, and error bounds. For bounded nonnegative factors, the convergents are monotone and converge to the least nonnegative solution in the admissible domain. Under the Favard identification, they approximate the mixed-type Weyl matrix. Unbounded factorizations retain the formal approximation; analytic convergence requires a closed realization and stability hypotheses, with an additional identification for measure-defined Weyl matrices. Piñeiro and Jacobi-like systems give applications, including global positive coefficientwise integrable solutions under the stated positivity and AT hypotheses.

math.CA↗