Papers

Empirical Bayes methods are among the most widely used statistical methods for large-scale inference. A central paradigm is the NPMLE, whose theoretical guarantees are by now well understood for the independent Gaussian sequence model. In this paper, we study empirical Bayes estimation from dependent observations in the Gaussian sequence model. We show that the maximum Composite Marginal Likelihood (CML) estimator, which ignores all correlations in the likelihood, converges in weighted Hellinger distance at the rate $n_*^{-1/2}$, where $n_*=n/\kappa_0$ is the `effective sample size' determined solely by the number of observations $n$ and the spectral radius $\kappa_0$ of the correlation matrix of the Gaussian observations. A complementary minimax lower bound shows that $n_*$ indeed serves as the right complexity measure, and that the CML estimator is nearly rate optimal under general dependence. We consider two concrete applications. In the first, we consider Bayesian linear regression, where the signal prior is estimated via CML applied to the least squares estimator. In the second, we consider the more challenging Bayesian nonlinear single-index model, where prior is estimated by CML applied to a one-step debiased gradient descent. In both applications, although the full likelihood landscape can be arbitrarily complicated and intractable, our CML method is facilitated by exploiting the high-dimensional distribution of the auxiliary statistics through a correlated Gaussian sequence model. The key ingredient in the proof of our results is a sharp local maximal inequality for the log composite marginal likelihood process under dependent Gaussian observations. In contrast to standard empirical process methods, we prove this inequality by leveraging a recent geometric Brascamp-Lieb inequality for Gaussian measures.

In 2009 and 2010, Blackburn and Shokrieh independently found that the discrete logarithm can be computed efficiently on the sandpile group of a graph, meaning that sandpile groups are not secure for cryptography. We generalize this problem to cokernels of matrices with entries in the ring of integers $\mathcal{O}_K$ of a number field $K$. When $K$ has nontrivial class group, the failure of the Euclidean algorithm in $\mathcal{O}_K$ is an obstacle to generalizing previous methods. For $M$ in $\mathrm{M}_{n\times m}(\mathcal{O}_K)$, we overcome this obstacle to efficiently compute discrete logarithms in $\mathrm{cok}(M) = \mathcal{O}_K^n/M\mathcal{O}_K^m$. In particular, we find an algorithm with time complexity $\tilde{O}((m+n)^{\omega+1})$, where $\omega$ is an exponent of matrix multiplication, to compute discrete logarithms in $\mathrm{cok}(M)$ when $\mathrm{cok}(M)$ is viewed either as an $\mathcal{O}_K$-module or as a group. When $M$ is Hermitian with respect to a Galois involution $\sigma$ and nonsingular, we improve the time complexity to $\tilde{O}(n^\omega)$.

Neyman–Pearson classification prioritizes one class by constraining its accuracy above a prespecified level, and then takes the accuracy of the other class as the utility objective. This paradigm is well suited for disease screening and diagnosis, among other applications. Statistical learning under this framework is complicated since classifier performance determines its acceptability. Furthermore, no learned classifier that is consistent for the oracle classifier can guarantee satisfaction of the control constraint in finite samples. Classical learning theory targets a control-relaxed empirical utility maximization (EUM) classifier. However, even the EUM classifier fails to achieve the desired control level on average. We conjecture that this under-control phenomenon is a manifestation of the over-optimism bias well known in standard statistical learning, and develop asymptotic theory to confirm it. Motivated by this insight, we propose refined learning procedures under two accuracy control strategies for the prioritized class: one controlling accuracy in expectation and the other with high probability. We further develop training-data-based methods to predict and infer class-specific accuracies of the resulting classifiers. Simulation studies demonstrate favorable finite-sample performance, and we illustrate the proposed methods with an application to cancer detection.

We settle a major case in the two-set regime of the Erdős similarity conjecture: the sum of a geometric sequence and an arbitrary infinite set is never measure universal. Here a set $E\subset\R$ is measure universal if every measurable set of positive Lebesgue measure contains an affine copy of $E$. More precisely, if $A\subset\R$ is infinite, $a\neq 0$, and $0<|r|<1$, then neither \[ \{ar^n:n\ge 1\}+A \qquad\text{nor}\qquad \{ar^n:n\ge 1\}-A \] is measure universal. More generally, the same conclusion holds when the geometric sequence is replaced by any set containing a lacunary sequence $(b_n)$ with $-\log b_n=O(n)$. Bourgain proved non-universality for sums of three arbitrary infinite sets, whereas the two-set regime is one of the principal remaining cases. Crucially, our conclusion applies to $\{2^{-n}\}+A$ for every infinite $A$, even though the non-universality of $\{2^{-n}\}$ itself remains open. The arbitrary-summand theorem is the maximal lacunary-density endpoint of a general counting-function trade-off. If $S_1,S_2\subset\R$ contain lacunary subsequences and $I(W),J(W)$ count their terms that are at least $e^{-W}$, then $S_1+S_2$ and $S_1-S_2$ are not measure universal whenever \[ \limsup_{W\to\infty}\frac{I(W)J(W)}{W}=\infty. \] No scale-separation or relative-decay assumption is required. The proof combines a finite-grid implementation of Kolountzakis' criterion with a near-additive-energy estimate controlling the clustering of lacunary cross-sums. A packing-number variant replaces lacunarity on one factor by a quantitative metric-mass condition. In particular, for $\alpha_1,\alpha_2>0$, the stretched-exponential sumset \[ \{2^{-n^{\alpha_1}}\}+\{2^{-n^{\alpha_2}}\} \] is not measure universal whenever $1/\alpha_1+1/\alpha_2>1$; analogous conclusions hold for difference sets.

Stechkin's classical results on the best lacunary system in $L_p$ spaces, given by the direction cosines on the unit sphere, are extended to Orlicz spaces $L_{\Phi}$. It is shown that for any $N$-function $\Phi$ the $L_{\Phi}$-norm of a linear combination of the direction cosines is completely determined by the $l_2$-norm of the coefficient vector. Consequently, the system is $S_{\Phi}(M)$-lacunary with a constant $M = K_{\Phi,n} \sqrt{n}$, where $K_{\Phi,n}$ coincides with the $L_{\Phi}$-norm of a single coordinate function. Moreover, under the additional convexity condition on $\Phi(\sqrt{u})$, this constant is proved to be optimal, so the direction cosines form the best lacunary system in Orlicz spaces. Explicit formulas for the constant are derived for $N$-functions $\Phi(u)=e^{u^2}-1$ and $\Phi(u)=\cosh u - 1$, and expressed in terms of hypergeometric functions.

We study generalizations of the classical Rogers–Shephard inequalities in the framework of Firey $L_p$-summation. We first consider the class of asymmetric $L_p$-zonoids. In this setting, we show that proving a sharp $L_p$-Rogers–Shephard inequality for asymmetric $L_p$-zonoids in $\mathbb{R}^n$ is equivalent to proving a sharp inequality between the volumes of projections of $B_q^m\cap \mathbb{R}^m_+$ and $B_q^m$ onto an $n$-dimensional subspace $E$, where $q$ is the Hölder conjugate of $p$. We conjecture that the inequality is sharp when the subspace $E$ is a coordinate subspace. We fully establish this inequality along with equality conditions in the case $p =2$. For general $p$, we prove it in the case $n=m-1$, $n=1$, and discuss several particular cases, including an averaged version and a local version of the inequality. We then turn to the setting of convex bodies having a center of symmetry. Rogers and Shephard also proved a sharp version of their inequality for bodies in this class. We conjecture a similar bound for the $L_p$-summation, and we establish our conjecture for the particular case of asymmetric $L_1$-zonoids, which, in particular, proves our conjecture in the planar case.

In 1997, Alon and Kim conjectured that if $H$ is a $k$-uniform $t$-simple hypergraph with maximum degree $D$ sufficiently large, then the chromatic index $\chi'(H)$ is upper bounded by $(t-1+1/t+\varepsilon)D$. Using probabilistic techniques and a nibble coloring method, we prove a general coloring theorem stating that a $k$-uniform $t$-simple hypergraph $H$ with large maximum degree $D$ satisfies $$\chi'(H) \le (b+\varepsilon)kD,$$ where $b$ is a particular parameter derived from local structural information about $H$. We use structural techniques to prove sharp upper bounds on $b$ in the 3-uniform 2-simple, and 3-uniform 3-simple cases. In particular, we deduce as a corollary that for sufficiently large $D$, every 3-uniform 2-simple and 3-simple hypergraph of maximum degree at most $D$ has chromatic index at most $2.3581D$ and $2.6791D$, respectively.

In this paper, we study integral operators \begin{equation*} T_\alpha f(x)=\int_{\mathbb{R}^{n}}K(x,y) f(y)dy, \end{equation*} with kernels $K(x,y)= k_1( x- A_1y)...k_m( x-A_my),$ where $k_i(x)=\frac{\Omega_i(x)}{|x|^{n/q_i}}$ and $\Omega_i: \mathbb{R}^n\to \mathbb{R}$ are homogeneous functions of degree zero, satisfying a size and a Dini condition, $A_{i}$ are certain invertible matrices, and $\frac n{q_1}+\dots+\frac n{q_m}=n-\alpha,$ $0\leq \alpha <n.$ We obtain the $H^{p}_{w^p}(\mathbb{R}^{n})-L^{q}_{w^q}(\mathbb{R}^{n})$ boundedness of these operators, for a class of Muckenhoupt weights $w$, satisfying the condition \begin{equation*} w(A_ix)\leq cw(x), \end{equation*} $a.e.x\in\mathbb{R} ^n $, $1\leq i\leq m$.

We prove the Eigenstate Thermalisation Hypothesis (ETH), also known as Quantum Unique Ergodicity (QUE), for large $N\times N$ mean-field random matrices with general correlation structure. We identify the microcanonical ensemble and establish the optimal fluctuation scale of eigenvector overlaps around it. Our results invalidate the inverse-density scaling predicted by Feingold and Peres [Phys. Rev. A 34, 591 (1986)] (and incorporated into Srednicki's ansatz [Phys. Rev. E 50, 888-901 (1994)]) in the physics literature of quantum chaos, based upon popular semiclassical theory, and uncover the genuine mechanism which relies on multi-resolvent local laws. Although fluctuations are expected to increase as the density of states vanishes, and indeed scale as $N^{-1/2}$ in the special cusp regime, rather than $N^{-1}$ in the bulk, we find, unexpectedly, that the same $N^{-1}$ rate persists at regular spectral edges. Hence, generically, in the absence of cusps, the entire eigenbasis fluctuates on the same scale as a Haar unitary. This anomaly stems from delicate cancellations in the solution of the underlying matrix Dyson equation, which form the core of our analysis.

For any finite set $\mathcal L$ of 3 or more skew lines in $\mathbb P^3_{\overline{K}}$ over an algebraically closed field $\overline{K}$ of arbitrary characteristic, there is a canonical associated subgroup $G_{\mathcal L}$ of ${\rm PGL}_2(\overline{K})$. Given a finite subgroup $G\subset{\rm PGL}_2(\overline{K})$ we study which configurations of lines have $G_{\mathcal L}=G$. We derive an upper bound on the number $|\mathcal L|$ of lines in terms of the order $|G|$ of the group $G$ and as an application we classify up to projective equivalence which sets $\mathcal L$ in $\mathbb P^3_{\mathbb C}$ have $G_{\mathcal L}=G$ for certain finite nonabelian groups $G$.

How do we define a bulk subsystem in quantum gravity? In \cite{Wong:2025kpz}, we argued that such a subsystem must support local holographic degrees of freedom. These are gravitational edge modes, whose entanglement creates a backreaction that fuses together subregions of spacetime. In this work we give a realization of these ideas in topological string theory, building upon \cite{Donnelly:2020teo,Jiang:2020cqo}. In this theory, a subsystem for closed strings consists of open strings ending on entanglement branes, which play the role of a dynamical entangling surface. Local holography is implemented by the geometric transition of these branes. We define a subregion open string algebra and develop a diagrammatics for open string modular flow for arbitrary states and subregion. We check that the entanglement entropy of these open strings reproduces the gravitational entropy of the associated closed string background. Finally, we relate these local transitions to defect holography.

The split-step-Padé (SSP) method is widely used to model wave phenomena in various applications, including radio physics, optics and acoustics. In this method, the propagator of the one-way counterpart of the Helmholtz equation is computed through its Padé approximant and a finite-difference discretization of the transverse operator. This work develops and validates numerically a spectral counterpart of the SSP method. A key challenge in practical applications is inverting the transverse operator in the presence of perfectly matched layers (PMLs), which are commonly used to truncate the computational domain. Such inversion can be accomplished using Krylov subspace methods, which converge rapidly, provided that a suitable preconditioner is used. We also study the analytical properties of the spectral SSP marching scheme under periodicity conditions in the transverse variable. We validate the newly developed spectral SSP method numerically in two realistic test scenarios from radio physics and underwater acoustics.

Adaptive token processing has emerged as a promising approach for improving the efficiency of semantic communication systems. However, existing semantic communication frameworks largely overlook token-level multiple access and the impact of semantic interference among simultaneously transmitted semantic tokens. In this paper, we propose Adaptive Token Selection and Token-Domain Multiple Access (ATS-ToDMA), a novel cross-modal semantic communication framework that jointly performs semantic token selection, interference-aware scheduling, and semantic-aware power allocation. The proposed framework introduces a Semantic Signal-to-Interference-plus-Noise Ratio (SSINR) metric that captures the combined effects of channel impairments and semantic interference arising from token similarity. A transformer-based scheduler is developed to allocate selected semantic tokens across token-domain transmission slots while mitigating both intra-modal and cross-modal semantic interference. To characterize the behavior of the proposed system, analytical bounds on semantic interference and feasible token occupancy are derived, together with a closed-form approximation for semantic-aware power allocation. Simulation results demonstrate significant gains in semantic throughput and semantic decoding accuracy while reducing aggregate semantic interference and transmit power compared with OMA, Semantic NOMA, Random-TS, and Greedy ATS benchmarks.

We show that projected Adam for online optimization with arbitrary moment decay parameters $\beta_1,\beta_2\in[0,1)$ can have average regret bounded away from zero. A similar result of Reddi-Kale-Kumar from 2018 required $\beta_1<\sqrt{\beta_2}$. Similar to their result, we use a three-periodic sequence of linear functions on $[-1,1]$ with slopes $c,-1,-1$, though we use $c$ slightly larger than $2$. This nonzero average regret result extends to Adam variants such as AdamW, RMSProp, NAdam, Adan, AdaMax, Muon, and to an i.i.d. variant of the three-periodic sequence of slopes for Adam.

When the standard of pointwise consistency for statistical inference – convergence to the truth in every possible state of the world – is provably unachievable, the usual responses are to change the inferential target or to strengthen background assumptions. This paper pursues a third: hold the inference problem fixed and identify the highest standard that remains achievable. I define a hierarchy of standards weaker than pointwise consistency, cast in topological terms, requiring convergence to the truth not everywhere but on a “large” set of probability measures. The main result is an impossibility theorem: for finite-precision tests, converging to the truth densely within each hypothesis already forces inconsistency on a comeager – “topologically almost all” – set of measures, whenever the two hypotheses are dense in their union. Distribution-free testing of conditional independence is one such case. Two further theorems characterize, in purely topological terms, exactly when each weaker standard is achievable, complementing Boeken et al.'s (2026) analysis of pointwise consistency.

This paper develops the matrix-valued analogue of the reflectionless and oracle framework for Jacobi operators. Starting from matrix-valued Weyl–Titchmarsh $m$-functions on the Siegel upper half-space, we study the distance-decreasing action of transfer matrices, matrix-valued harmonic measures and value distribution convergence. These ingredients are then used to establish the reflectionless property of the $\omega$-limit set and to prove an Oracle Theorem for matrix-valued Jacobi operators with absolutely continuous spectrum of full multiplicity.

We show that the standard Veronese degree of the defining ideal $I_{n,h}$ of a monomial star configuration is $\lcm(1,2,\ldots,h)$, proving a conjecture by Grifo and Seceleanu. We also provide a counter-example for their question regarding upper bounds for the standard Veronese degree of monomial ideals.

We consider a swarm array of autonomous relays that seek to cooperatively forward a desired signal to a fusion center with the maximum possible fidelity while canceling out a number of interferers. We present a distributed algorithm for computing the optimal zero-forcing beamforming weights at the relays without requiring prior channel knowledge. Crucially, our algorithm is scale-free in the sense that the computational and bandwidth overheads are completely independent of the size of the array. We build on recent work that introduced the concept of a Collective Array that enables such scale-free computation by imposing a constraint that the array must always function as a swarm i.e. array elements can only ever communicate with external nodes collectively and never individually. While this is a very severe restriction, we show that it allows useful computations such as zero-forcing beamforming while being robust to noise and channel time-variations.

We present a method for computing reducible, normally hyperbolic, invariant tori with internal quasiperiodic dynamics in autonomous ordinary differential equation systems. The approach is based on the parameterization method of KAM theory; thus, it is a Newton scheme with small divisors. Since the inner dynamics of the torus is prescribed, the corresponding system parameters for which such a torus exists are simultaneously determined. The method is amenable to a form of pseudo-arclength continuation, enabling the traversal and computation of saddle-node bifurcations. We give explicit algorithms for the methods and demonstrate their applicability with two numerical examples.

We prove the existence of stably degenerate hypersurface singularities. Ingredients in the proof include the mixed Hodge structure on the contact locus of a singularity and that moduli spaces of polarised abelian varieties of large dimension are of general type.

We study a time-dependent non-Hermitian generalisation of the Schütte-Da~Providência model describing a bosonic mode coupled to collective particle-hole excitations. Using a time-dependent Dyson map, we construct a Hermitian counterpart and reduce the collective fermionic sector by means of the random phase approximation (RPA). The resulting dynamics is mapped to two time-dependent harmonic-oscillator branches with instantaneous RPA frequencies $W_\pm(t)$. We determine the corresponding stability regions and compute transition probabilities between instantaneous oscillator states. In first-order instantaneous-basis perturbation theory the leading transition $n\to n+2$ is proportional to $\dot W_j/W_j$, showing that it is purely nonadiabatic and absent in the time-independent case. We compare this result with exact Lewis-Riesenfeld transition amplitudes within the RPA approximation. Numerical examples show that different components of the Dyson map provide distinct driving mechanisms: the scaling parameter modulates the effective coupling, while the squeezing parameter acts through a moving-boundary contribution. In both cases the induced collective transitions exhibit parametric-resonance peaks and sideband structures.

Around 1910, in an unpublished manuscript, Ramanujan proposed the following identity for $\zeta(2n+1)$: \[ \begin{aligned} \alpha^{-n}\,&\left\{\dfrac{1}{2}\,\zeta(2n + 1) + \sum_{m = 1}^{\infty}\dfrac{m^{-2n - 1}}{e^{2\alpha m} - 1}\right\} \\ &\quad\quad\quad\quad\quad\quad\quad-(-\beta)^{-n}\,\left\{\dfrac{1}{2}\,\zeta(2n + 1) + \sum_{m = 1}^{\infty}\dfrac{m^{-2n - 1}}{e^{2\beta m} - 1}\right\}\\ &=2^{2n}\sum_{k = 0}^{n + 1}\dfrac{(-1)^{k-1}B_{2k}\,B_{2n - 2k + 2}}{(2k)!(2n - 2k + 2)!}\,\alpha^{n - k + 1}\beta^k, \end{aligned} \] where $\alpha$, $\beta$ are positive numbers satisfying $\alpha\beta=\pi^2,n\in\mathbb{N},$ $B_n$ denotes the $n$-th Bernoulli number and $\zeta(z)$ is the Riemann zeta function. As shown by Berndt in the viewpoint of general transformation of analytic Eisenstein series, it is a natural companion of Euler's famous formula for even zeta values. In this paper, we extend Ramanujan's identity to the alternating Hurwitz zeta function. Then we systematically investigate the properties of the alternating Hurwitz zeta function $\zeta_E(z,x)$, as well as the corresponding Ramanujan-type identities, under different modular symmetry conditions. We also establish infinite series expressions for products of the tangent and hyperbolic tangent functions, and express the Dirichlet lambda function $\lambda(z)$ together with linear combinations of infinite series as convolution sums of special sequences. Furthermore, we define alternating Hurwitz kernels of even and odd orders, and obtain Ramanujan-type identities involving the alternating digamma function $\widetilde{\psi}(x)$ and Euler polynomials $E_n(x)$, as well as transformation formulas between even-order and odd-order alternating Hurwitz kernels.

Mutual information estimation is a central problem in machine learning and statistics; however, existing benchmarks typically evaluate estimators on simplified, low-dimensional distributions, leaving their performance on complex, realistic data largely unexplored. We address this gap with a comprehensive benchmarking framework grounded in a unified copula-theoretic perspective that subsumes existing benchmarks as special cases. Within this framework, we propose two complementary families of tests: a copula-first family that systematically varies ground-truth mutual information, dimensionality, and marginal complexity using synthetic and flow-based transformations; and a marginals-first family that couples real-world image data with controlled dependency structures, extending the classic same-class-pairing paradigm. We use this suite to extensively evaluate three classes of estimators: non-parametric, variational, and generative. Contrary to prevailing assumptions, our results indicate that there is no universal winner: each category can systematically outperform all other estimators under specific setups. By analyzing these cases, we identify fundamental estimation barriers and propose new tests that more effectively stress these specific limitations. We share the open source code at https://github.com/VanessB/mutinfo.

Homotopic distance is a numerical homotopy invariant that quantifies the homotopic distinction between two or more continuous maps. In this paper, we look at various versions of homotopic distance between maps and study them on maps to group-like spaces and CW H-spaces. In particular, (1) we develop the theories of probabilistic and diagonalized versions of the homotopic distance and compare their properties, and (2) we show that all versions of the homotopic distance can be calculated precisely in terms of their respective versions of the Lusternik-Schnirelmann category when group-like spaces or CW H-spaces are involved. Our results are refined in the setting of rational groups. On the way, we also study loopings of maps and the nilpotency of the set of homotopy classes of maps to group-like spaces.

The purpose of this study is to determine the role of the Academy of Gondishapur in the formation of the Bayt al-Hikma (House of Wisdom) in Baghdad and the emergence of Islamic science in the 8th-10th centuries. The methodology combines comparative-historical and biographical analysis with a study of institutional continuity based on primary sources and contemporary scholarly literature. The results demonstrate that Gondishapur, the foremost scientific and medical centre of late Sasanian Iran, constituted the primary human, methodological, and institutional source for the Abbasid Bayt al-Hikma. The migration of specialists, especially the Bukhtishu dynasty, Yuhanna ibn Masawayh, and Hunayn ibn Ishaq, transferred Gondishapur's medical traditions, hospital organisation (bimaristans), and translation methodology to Baghdad. Gondishapur's experience also shaped the development of medical education and the compilation of pharmacopoeias within the Bayt al-Hikma. A comparative structural analysis of the two institutions reveals both institutional borrowings and the original features of the Baghdad model. The article concludes that the transmission of Gondishapur's heritage to the Bayt al-Hikma constituted sustained institutional continuity, laying the foundations for the large-scale synthesis of Greek, Persian, and Indian knowledge in the Arabic language.

The connections between graph theory, and more generally association schemes, and coding theory were established by Delsarte for the Hamming metric and rank-metric codes. The ambient metric space of Hamming-metric codes and rank-metric codes can be seen as Cayley graphs generated by words of weight one. The metrics considered then coincide with the geodesic distances of these distance-regular graphs. We focus on a generalisation of this framework to the space of tensors over a finite field, endowed with the tensor-rank as a metric. This space corresponds to the Cayley graph generated by rank-one tensors, which is not distance-regular for tensors of order at least 3. We show that the spectrum of this graph has a recursive expression and depends on the possible intersections between tensor subspaces of large enough dimension and the Segre variety. The spectrum of this graph for 3-order tensors can be expressed with the rank distribution of the rank-metric codes generated by these tensors. In particular, we obtain the complete spectrum of the graph for 2x3x3 tensors over any finite field. We apply this result to derive bounds on the dimension of tensor codes in the tensor-rank metric using the eigenvalue method, and in particular the ratio-type bound.

In this article, we establish the $A$-Berezin number and $A$-Berezin norm inequalities for bounded linear operators on a reproducing kernel Hilbert space using the Moore-Penrose inverse. We further extend these inequalities to the case of the sum of two bounded linear operators. As an application, upper bounds of the Berezin number and Berezin norm of block matrices have been discussed via the Moore-Penrose inverse. The inequalities established here offer both refinements and generalizations of previous results.

We solve a long-standing open problem, posed by Landweber in 1974: Let $R = K[x_1, x_2, . . . ]$ be the ring of polynomials in countably many variables over a field $K$. Is the formal power series ring $R[[t]]$ a unique factorization domain? We prove that it is. The proof is based on a new general result in commutative algebra: If $R$ is a Krull domain, and $f \in R[[t]]$ is irreducible, then $f$ is irreducible modulo a finite power of $t$.

In this paper, we study Anderson localization near the spectral edge for the Anderson-Bernoulli model on $\mathbb{Z}$ with long-range hopping. When the hopping has a rational Laurent symbol, a quantitative version of the unique continuation principle can be proved, and localization occurs. For the unique continuation in the general case, we give some counterexamples and prove a weaker result for hopping that decays faster than exponential rate. To the best of our knowledge, this is the first localization result for the long-range Anderson model with pure Bernoulli potentials.

Using a recently-introduced quantum control model for Unruh-DeWitt detectors in superpositions of classical trajectories, we investigate the response of a detector interacting with a massless scalar quantum field in (3+1) dimensions along a superposition of circular trajectories. We present numerical results for the transition probability and effective temperature of such a detector in four distinct geometric scenarios: (a) concentric, vertically-stacked trajectories, (b) planar, horizontally-displaced trajectories, (c) static central point and surrounding circular trajectory, and (d) concentric, planar circular trajectories. For Gaussian switching functions that are much broader than the acceleration timescale, in case (a) we find only minor deviations from the well-known, effectively thermal response of a single circular trajectory, whereas in case (c) we find a significant reduction in the effective temperature and greater variation with energy gap. We conclude with a discussion of a potential analogue implementation in ultracold atom systems.