The eigenvalue problem for a probability measure $\mu$ with compact support in $\R$ is whether there exist a countable set $\Lambda$ and a nonzero real $t\ne 1$ such that both $\Lambda$ and $t\Lambda$ are spectra of $\mu$, that is, the family $$E_{a\Lambda}=\{e^{-2\pi i a\lambda x}:\lambda\in\Lambda\}$$ is an orthonormal base for $L^2(\mu)$ for $a=1, t$. The eigenvalue problem was discovered independently by Strichartz (Str00), Laba and Wang (LW02) for the Cantor measures $\mu_{4,\{0,1\}}$ and $\mu_{6,\{0,1,2\}}$, respectively. In this paper, we investigate the spectral eigenvalue problem for the general spectral Cantor measure $\mu_{b,\mathcal{D}}$. 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 (Min57).
Let $\Lambda \subset \mathbb{R}^n$ be a full rank lattice with the property that $\det \Gamma \ge 1$ for all sublattices $\Gamma \subseteq \Lambda^*$. We prove that there exists a symmetric random vector $X \in \mathbb{R}^n$ with a smooth, strictly positive density $f : \mathbb{R}^n \to (0, \infty)$ satisfying that $f$...
Brayden Letwin, Victor Reis, T. Rothvoss· 0 citations
Let $\Phi$ be an irreducible root system normalized so that long roots have length $\sqrt{2}$, and let $\mu\leq\lambda$ be dominant weights. We associate to $(\Phi,\mu,\lambda)$ the convex polytope $$P^{\Phi}_{\mu,\lambda} = (\mu+C)\cap(\lambda-C)\cap(\overline{D}-\rho),$$ where $C=\mathbb{R}_{\geq 0}\Phi^+$ and $\over...
Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with $n\geq2$. The well-known Payne-P\'olya-Weinberger conjecture for consecutive eigenvalue ratios asserts that $$ \frac{\lambda_{m+1}(\Omega)}{\lambda_m(\Omega)}\leq\left(\frac{j_{n / 2,1}}{j_{n / 2-1,1}}\right)^2,\quad m=1,2,3,\cdots. $$ Here, $j_{p,k}$ denotes the...
Let $\mathcal{S}^ r$ denote the Schatten--von Neumann class and let $S_{\Psi_{f,\lambda}}$ be the Schur--Hadamard multiplier whose symbol is the divided-difference matrix of $f$ along $\lambda$. Let $0<\alpha,r<\infty$, let $f\in C^1([-1,1])$ satisfy $f(0)=0$, $|f'(t)|\lesssim |t|^\alpha$, and let $\lambda\in\ell^r$ be...
Let $\mathcal{H}_k$ denote the set of normalized holomorphic Hecke cusp forms of weight $k$ for the full modular group $ \mathrm{SL}_2(\mathbb Z)$. For each $f\in\mathcal{H}_k$, let $\lambda_f(n)$ denote the corresponding eigenvalue of the normalized Hecke operator $T_n$. We prove nontrivial upper bounds for \[\sum_{f...
Let $(M,g)$ be a compact $d$-dimensional Riemannian manifold without boundary, and let $\{e_j\}_{j=0}^\infty$ be an orthonormal basis of Laplace eigenfunctions with frequencies $\{\lambda_j\}_{j=0}^\infty$. For a domain $\Omega\subset M$, consider $$ N_\Omega(\lambda) = \sum_{\lambda_j\leq\lambda} \left|\int_\Omega e_j...
Xu-Cheng Hu, Ya-Kun Xi· 0 citations
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