The periodic tiling conjecture asserts that if a bounded set $Ω\subset \mathbb R^d$ of finite measure tiles $\mathbb R^d$ by translations, then it admits at least one fully periodic tiling. This conjecture was proved to hold in $\mathbb R$ by Lagarias and Wang, and was recentl...
We prove that for every window $w$ of the form $w(x) = e^{-π|x|^2} h(x)$, where $h$ is a polynomial, there exists a uniformly discrete set of points $\mathcal{S} \subset \mathbb R^{2d}$ such that the magnitude of the short-time Fourier transform $V_w f$ on $\mathcal{S}$ determ...
We prove that the set of time-frequency shifts $\{e^{2πi βl t} g(t-αk) : k,l \in \mathbb{Z}\}$ with a continuous, integrable totally positive function $g$ and lattice parameters $α,β>0$ generates a frame for $L^2(\mathbb{R})$ if and only if $αβ<1$. This fully settles the so-ca...
The Schiffer conjecture states that if a smooth domain $Ω\subset \mathbb{R}^n$ admits a Neumann eigenfunction of the Laplacian which is constant at the boundary, then the domain is a ball. It is intimately related to Pompeiu's problem, stating that if a nonzero function integr...
The ability of large language models to assist professional mathematicians has been progressing rapidly. Earlier this year, a group of researchers in Banach lattice theory and phase retrieval began incorporating this technology into their research workflows. Facing challenges...
We prove that, after L2 normalization, stable phase retrieval holds over the L2-spans of independent real-valued centered random variables if and only if all but possibly one coordinate satisfies a uniform two-sided L1 bound. This provides a complete characterization of stable...
We prove that the Cantor measure with base b does not admit a Fourier frame whenever b>1 is an odd integer. In particular, this answers a question of Strichartz on the existence of a Fourier frame for the middle third Cantor measure. A formalization of our main result in Lean...
For every dimension d>1, we establish explicit criteria on lattices Λ⊂ℝ2d with density D(Λ)>1 such that no function with a continuous Zak transform generates a Gabor frame along Λ. In particular, this gives a negative answer to the existence problem of Gabor frames with window...
We study the squared singular value spectrum of a product of non-square random matrices, a setting that also corresponds to the feature covariance eigenvalues of a deep linear neural network at initialization. We first take a proportional depth-width d,n limit with the number...
Carbery proposed the following sharpened form of triangle inequality for many functions: for any $p\ge 2$ and any finite sequence $(f_j)_j\subset L^p$ we have \[ \Big\|\sum_j f_j\Big\|_p \ \le\ \left(\sup_{j} \sum_{k} α_{jk}^{\,c}\right)^{1/p'} \Big(\sum_j \|f_j\|_p^p\Big)^{1/...
We prove that the short-time Fourier transform with Gaussian window performs L2-local stable phase retrieval at the constant function. The proof involved significant interplay between mathematicians and LLMs. An autoformalization in Lean 4 of an extension of our result to L2-l...
The hot spots ratio of a domain Ω⊂Rd measures the degree of failure of Rauch's hot spots conjecture on that domain. We identify the largest possible value of this ratio over all connected Lipschitz domains Ω⊂Rd, for any dimension d. As d→∞, we show that this maximal ratio conv...
The hot spots conjecture asserts that for any convex bounded domain $\Omega$ in $\mathbb{R}^d$, the first non-trivial Neumann eigenfunction of the Laplace operator in $\Omega$ attains its maximum at the boundary. We construct counterexamples to the conjecture for all sufficien...
We investigate the problem of predicting the output behavior of unknown quantum channels. Given query access to an n-qubit channel E and an observable O, we aim to learn the mapping ρ↦Tr(OE[ρ]) to within a small error for most ρ sampled from a distribution D. Previously, Huang...
The periodic tiling conjecture asserts that if a region Σ⊂Rd tiles Rd by translations then it admits at least one fully periodic tiling. This conjecture is known to hold in R, and recently it was disproved in sufficiently high dimensions. In this paper, we study the periodic t...
Log-concave sampling has witnessed remarkable algorithmic advances in recent years, but the corresponding problem of proving lower bounds for this task has remained elusive, with lower bounds previously known only in dimension one. In this work, we establish the following quer...
In this paper we prove a uniform Fourier restriction estimate over the class of simple curves where the last coordinate function can be extended to a holomorphic function of bounded frequency in a sufficiently large disc. The proof is based on a decomposition scheme for this c...
We give new proofs of the description convex hulls of space curves $\gamma : [a,b] \mapsto \mathbb{R}^{d}$ having totally positive torsion. These are curves such that all the leading principal minors of $d\times d$ matrix $(\gamma', \gamma'', \ldots, \gamma^{(d)})$ are positiv...
We prove that for $d\geq 0$ and $k\geq 2$, for any subset $A$ of a discrete cube ${0,1}^d$, the $k-$higher energy of $A$ (i.e., the number of $2k-$tuples $(a_1,a_2,\dots,a_{2k})$ in $A^{2k}$ with $a_1-a_2=a_3-a_4=\dots=a_{2k-1}-a_{2k}$) is at most $|A|^{\log_{2}(2^k+2)}$, and...
In this paper we study an autocorrelation inequality proposed by Barnard and Steinerberger. The study of these problems is motivated by a classical problem in additive combinatorics. We establish the existence of extremizers to this inequality, for a general class of weights,...
We consider decoupling for a fractal subset of the parabola. We reduce studying l2Lp decoupling for a fractal subset on the parabola {(t,t2):0≤t≤1} to studying l2Lp/3 decoupling for the projection of this subset to the interval [0,1]. This generalizes the decoupling theorem of...
The analysis of neural network training beyond their linearization regime remains an outstanding open question, even in the simplest setup of a single hidden-layer. The limit of infinitely wide networks provides an appealing route forward through the mean-field perspective, bu...
The aim of this paper is to prove a uniform Fourier restriction estimate for certain 2−dimensional surfaces in R2n. These surfaces are the image of complex polynomial curves, equipped with the complex equivalent to the affine arclength measure. This result is a complex-polynom...
Urban spatial networks are complex systems with interdependent roles of neighborhoods and methods of transportation between them. In this paper, we classify docking stations in bicycle-sharing networks to gain insight into the spatial delineations of three major United States...