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We address a maximally structured case of the question, "Can you hear your location on a manifold," posed in arXiv:2304.04659 for dimension $2$. In short, we show that if a compact surface without boundary sounds the same at every point, then the surface has a transitive action by the isometry group. In the process, we show that you can hear your location on Klein bottles and that you can hear the lengths and multiplicities of looping geodesics on compact hyperbolic quotients., 9 pages, 1 figure

We introduce a variation on Kac's question, "Can one hear the shape of a drum?" Instead of trying to identify a compact manifold and its metric via its Laplace--Beltrami spectrum, we ask if it is possible to uniquely identify a point $x$ on the manifold, up to symmetry, from its pointwise counting function \[ N_x(\lambda) = \sum_{\lambda_j \leq \lambda} |e_j(x)|^2, \] where here $\Delta_g e_j = -\lambda_j^2 e_j$ and $e_j$ form an orthonormal basis for $L^2(M)$. This problem has a physical interpretation. You are placed at an arbitrary location in a familiar room with your eyes closed. Can you identify your location in the room by clapping your hands once and listening to the resulting echos and reverberations? The main result of this paper provides an affirmative answer to this question for a generic class of metrics. We also probe the problem with a variety of simple examples, highlighting along the way helpful geometric invariants that can be pulled out of the pointwise counting function $N_x$., Comment: 26 pages, 1 figure

We study the Vapnik-Chervonenkis (VC) dimension of the set of quadratic residues (i.e. squares) in finite fields, $\mathbb F_q$, when considered as a subset of the additive group. We conjecture that as $q \to \infty$, the squares have the maximum possible VC-dimension, viz. $(1+o(1))\log_2 q$. We prove, using the Weil bound for multiplicative character sums, that the VC-dimension is $\geq (\frac{1}{2} + o(1))\log_2 q$. We also provide numerical evidence for our conjectures. The results generalize to multiplicative subgroups $\Gamma \subseteq \mathbb F_q^\times$ of bounded index., Comment: 21 pages, 3 figures

Consider an $L^2$-normalized Laplace-Beltrami eigenfunction $e_\lambda$ on a compact, boundary-less Riemannian manifold with $\Delta e_\lambda = -\lambda^2 e_\lambda$. We study eigenfunction triple products \[ \langle e_\lambda e_\mu, e_\nu \rangle = \int e_\lambda e_\mu \overline{e_\nu} \, dV. \] We show the overall $\ell^2$-concentration of these triple products is determined by the measure of some set of configurations of triangles with side lengths equal to the frequencies $\lambda,\mu,$ and $\nu$. A rapidly vanishing proportion of this mass lies in the `classically forbidden' regime where $\lambda, \mu,$ and $\nu$ fail to satisfy the triangle inequality. As a consequence, we improve a result by Lu, Sogge, and Steinerberger., Comment: 29 pages, 1 figure

Let $(M,g)$ be a compact, 2-dimensional Riemannian manifold with nonpositive sectional curvature. Let $\Delta_g$ be the Laplace-Beltrami operator corresponding to the metric $g$ on $M$, and let $e_\lambda$ be $L^2$-normalized eigenfunctions of $\Delta_g$ with eigenvalue $\lambda$, i.e. \[ -\Delta_g e_\lambda = \lambda^2 e_\lambda. \] We prove \[ \left| \int_{\mathbb R} b(t) e_\lambda (\gamma(t)) \, dt \right| = o(1) \quad \text{ as } \lambda \to \infty \] where $b$ is a smooth, compactly supported function on $\mathbb R$ and $\gamma$ is a curve parametrized by arc-length whose geodesic curvature $\kappa(\gamma(t))$ avoids two critical curvatures $\mathbf k(\gamma'^\perp(t))$ and $\mathbf k(-\gamma'^{\perp}(t))$ for each $t \in \operatorname{supp} b$. $\mathbf k(v)$ denotes the curvature of a circle with center taken to infinity along the geodesic ray in direction $-v$.