1,914 papers
2604.04799On a $_2F_1\big(\frac{1}{4}\big)$-identity due to Gosper
It is only in exceptional cases that a $_2F_1(z)$-series with rational parameters and a rational argument, apart from the cases for $z \in \{ \pm 1, \frac{1}{2} \}$ associated with classical hypergeometric identities, admits an evaluation given by a combination of $Γ$-values with rational arguments. In this paper, we present a new and integration-based approach toward the construction of special values for $_2F_1$-series of the desired form. We apply this approach using a $_2F_1\big(\frac{1}{4}\big)$-identity originally due to Gosper and later considered by Vidunas, Ebisu, and Zudilin, to evaluate a ${}_{2}F_{1}$-series of convergence rate $\big(\frac{172872}{185039}\big)^2$. With regard to extant research on so-called ``strange'' ${}_{2}F_{1}$-evaluations, as in the work of Ebisu and Zeilberger, our new series seems to have the largest numerator/denominator in its argument.
2604.04799Apr 2026
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From hyperbolic to complex Euler integrals
Hyperbolic hypergeometric integrals are defined as Barnes-type integrals of products of hyperbolic gamma functions. Their reduction to ordinary hypergeometric functions is well known. We study in detail their degeneration to complex hypergeometric functions. Namely, using uniform bounds on the integrands, we prove that the univariate hyperbolic beta integral and the conical function degenerate to two-dimensional integrals over the complex plane.
2604.04471Apr 2026
View2604.04268Bernstein inequality on parabolic domains
Several families of sharp Bernstein inequalities are established on the weighted $L^2$ space over parabolic domains, which include bounded or unbounded rotational paraboloids and parabolic surfaces. The main tool is a second-order differential operator satisfied by a specific basis of orthogonal polynomials in weighted $L^2$ space.
2604.04268Apr 2026
View2604.03680Separating zeros of polynomials using an added interlacing point
Following a systematic analysis of existing results, we investigate when complete interlacing between the zeros of distinct polynomial sequences, $\{\mathcal{P}_n\}$ and $\{\mathcal{G}_n\}$ can be achieved by using a naturally arising extra point. Specifically, we analyse several general mixed recurrence relations that ensure the $n+1$ zeros of the polynomial $(x-E)\mathcal{P}_n(x)$ interlace with the $k$ zeros of $\mathcal{G}_k$, where $k=n$ or $n+1$. In addition, we show that imposing specific conditions on the extra point $E$ yields full interlacing between the zeros of $\mathcal{P}_n$ and $\mathcal{G}_k$ for a suitable choice of $n$. The approach provides a consolidated framework broadly applicable to both orthogonal and non-orthogonal polynomials and we illustrate this with new interlacing results for zeros of Krawtchouk, Meixner, and Narayana polynomials. We also illustrate that this general approach can be used to recover and refine existing results regarding the complete interlacing of zeros for classical Jacobi and Laguerre polynomials.
2604.03680Apr 2026
View2604.03638Extension theorems for logarithmic Schrödinger and discrete Laplacian operators
In this paper we consider logarithmic operators in two different contexts: the adapted to (continuous) Schrödinger operators and the classical discrete setting. The Schrödinger operator $\mathcal L_V$ on $\mathbb R^d$ is defined as $\mathcal L_V=-Δ+V$, where the potential $V$ is nonnegative and satisfies a reverse Hölder inequality and, as usual, $Δ$ denotes the Euclidean Laplacian, while the discrete Laplacian $Δ_d$ on $\mathbb Z$ is given by $(Δ_df)(n)=f(n+1)-2f(n)+f(n-1)$, $n\in \mathbb Z$. Both logarithmic operators $\log \mathcal L_V$ and $\log (-Δ_d)$ are nonlocal operators and we will define them through suitable extension problems. The extension problems for logarithmic operators are inspired by the one introduced by Caffarelli and Silvestre for the fractional Laplacian but, in this case, the logarithmic operators are obtained as the boundary values of the extension in a more involved way.
2604.03638Apr 2026
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The Kakeya conjecture, after Wang and Zahl
This is a Seminaire Bourbaki survey of the proof of the Kakeya conjecture in three dimensions. The survey is written for a broad mathematical audience. We sketch all the ideas in the proof, with many pictures.
2604.03416Apr 2026
View2604.02984A Bilinear Kakeya Inequality in the Heisenberg Group
We prove a bilinear Kakeya inequality in the first Heisenberg group and a sharp bilinear Kakeya estimate for Euclidean curved tubes in $\R^2$. By adapting an argument of Fässler, Pinamonti and Wald involving Heisenberg projections, we show that the latter implies the former. We prove the estimate for curved tubes using a combination of techniques developed by Pramanik, Yang and Zahl, Wolff and Schlag. We introduce a novel broadness hypothesis inspired by works of Zahl, which rules out bush-type configurations that break transversal structure. We argue that such a hypothesis is needed for proving the bilinear estimates we present. We also introduce necessary additional linear terms to the estimate to counteract Szemerédi--Trotter-type clustering phenomena.
2604.02984Apr 2026
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Cesàro summability of Hölder functions and Talbot effect on rank one Riemannian symmetric spaces of compact type
On rank one Riemannian symmetric spaces of compact type (of dimension $\ge 2$), we first obtain a quantitative characterization of Hölder continuity in terms of Cesàro means. In addition to some approximation theoretic applications, we also apply it to study the celebrated physical phenomenon known as `Talbot effect' arising from diffraction theory. More precisely, for almost every fixed time instance, we study the Hölder continuity and the fractal profile of the Schrödinger propagation in terms of the decay of the Littlewood-Paley projections of the initial data. In the process, we also obtain oscillatory expansions of zonal spherical functions uniformly near the origin and near the cut locus respectively, which may be of independent interest.
2604.02305Apr 2026
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A Fenchel Theorem for the Gauss maps and uniqueness of minimizers of nonlocal curvature energies
In this paper, we prove a Fenchel theorem for Gauss maps by providing sharp lower bounds for the path length of Gauss maps of an embedding. By combining the Fenchel-type theorem with various techniques from the field of geometric analysis, we show that circles minimize most generalized tangent-point energies. Furthermore, we prove that disks minimize all fractional Willmore energies among the class of convex planar sets.
2604.02042Apr 2026
View2604.01998Systems with discrete singular $φ$-Laplacian and maximal monotone boundary conditions
We are concerned with solvability of nonlinear systems involving a discrete singular $φ$-Laplacian operator of type \begin{equation*} u \mapsto Δ\left[φ(Δu(n-1))\right] \qquad (n\in \{1, \dots, T\}), \end{equation*} associated with a general two point boundary condition having the form \begin{equation*} \left(φ(Δu(0)),-φ(Δu(T))\right)\inγ(u(0),u(T+1)), \end{equation*} where $γ:\mathbb{R}^N\times\mathbb{R}^N\to2^{\mathbb{R}^N\times\mathbb{R}^N}$ is a maximal monotone operator with $0_{\mathbb{R}^N \times \mathbb{R}^N}\in γ(0_{\mathbb{R}^N \times \mathbb{R}^N})$. The mapping $φ$ is a potential homeomorphism from an open ball of radius $a$ centered at the origin $B_a \subset \mathbb{R}^N$ onto $\mathbb{R}^N$ and $Δ$ stands for the usual forward difference operator. When the perturbing nonlinearity in the system has not a potential structure we obtain existence of solutions by a priori estimates. Also, when the nonlinearity is of gradient type and $γ$ is a subdifferential, we provide a variational approach of the system in the frame of critical point theory for convex, lower semicontinuous perturbations of $C^1$-functionals. Then we derive the existence of solutions either as minimizers or saddle points of the corresponding energy functional.
2604.01998Apr 2026
View2604.01525A Determinantal Approach to a Sharp $\ell^1-\ell^\infty-\ell^2$ Norm Inequality
We give a short linear--algebraic proof of the inequality \[ \|x\|_1\,\|x\|_\infty \le \frac{1+\sqrt{p}}{2}\,\|x\|_2^2, \] valid for every \(x\in\mathbb{R}^p\). This inequality relates three fundamental norms on finite-dimensional spaces and has applications in optimization and numerical analysis. Our proof exploits the determinantal structure of a parametrized family of quadratic forms, and we show the constant $(1+\sqrt{p})/2$ is optimal.
2604.01525Apr 2026
View2604.01493Falconer lattice sets and the Erdos similarity problem
We show that a family of extremely thin sets satisfy the Erdős similarity conjecture. These examples lie outside the range covered by recent work of Shmerkin and Yavicoli \cite{ShmerkinYavicoli2025}. As we shall see, they have small logarithmic dimension. They do not contain affine copies of slowly decaying sequences, so the result does not follow from earlier work of Falconer and Eigen \cite{Falconer1984,Eigen}. On the other hand, they do contain sequences of rapid decay, for which the conjecture is still open in general. Our argument is based on Falconer lattice sets and a theorem of Bourgain \cite{Bourgain2003}.
2604.01493Apr 2026
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Extremal graph theory and point configurations in Ahlfors-David regular sets
We study the problem of embedding bipartite graphs in Ahlfors-David regular sets of large dimension using results from extremal graph theory. Our main theorem states that any graph satisfying a power-improving bound on the extremal number can be found in the distance graph of a sufficiently high-dimensional AD-regular set. In particular, we show that AD-regular sets of dimension greater than $\frac{d+1}{2}$ must contain even cycles of all lengths if $d\geq 3$, and must contain even cycles of length at least 6 if $d=2$. This improves the best known threshold for the problem in $d\geq 4$, and yields entirely new results in $d=2,3$, under the extra assumption of AD-regularity. We also prove analogous results for large subsets of vector spaces over finite fields, which improve the best known exponent for even cycles in all dimensions.
2604.01381Apr 2026
View2604.00714Characterizations of fractional operators via integral transforms
In 1972, J. S. Lew established a reasonable conjecture regarding an axiomatic characterization for the one-dimensional Riemann-Liouville integral. This conjecture was proved by Cartwright and McMullen in 1978. After that, little further work has been done on this topic, except some extensions for the Stieltjes case in one and several variables. In this paper, we prove the necessity of the axioms established in the conjecture of J. S. Lew using the Cauchy functional equation and Hamel bases. In addition, we give a proof for the characterization in several variables by employing Titchmarsh theorem, as a natural extension of the approach of Cartwright and McMullen. We also provide an alternative version and proof in one and several variables with Laplace transforms and the Cauchy functional equation, weakening parts of the continuity assumption. We show a similar result for the Riesz potential in terms of the Fourier transform. Finally, we illustrate how the theory can be used for characterization in the context of fractional calculus with respect to a non-smooth integrator, based on transmutation and measures.
2604.00714Apr 2026
View2604.00345Maximal functions with twisted structures, distribution inequality and applications
Motivated by the geometric reduction of Cauchy--Szegő projections on quadratic surfaces of higher codimension (Nagel--Ricci--Stein, 2001) and recent developments on the real-variable theory adapted to twisted multiparameter structures (arXiv:2603.26119), we establish the Fefferman--Stein type distribution inequality relating the twisted area function and the twisted non-tangential maximal function over $\mathbb{R}^{2m}$. By deploying a recursive integration-by-parts argument involving the twisted gradient and Laplacian, and constructing smooth, compactly supported weight functions to absorb cross-derivative errors, we obtain the required estimate. As an application, we prove the uniform $L^1$ boundedness of the twisted maximal function on the twisted atoms and complete the maximal function characterization of the twisted Hardy space.
2604.00345Apr 2026
View2603.29600Uniform optimal-order Wasserstein quantisation
We address Steinerberger's Wasserstein transport problem on the cube $Q=[0,1]^d$. For every $d\ge2$, we consider a dyadic digital sequence $(x_n)\subset Q$ and prove that every prefix $\{x_1,\dots,x_N\}$ admits an exact equal-mass transport partition at the optimal scale. More precisely, for every $N\in\mathbb{N}$, there exist pairwise disjoint Borel sets $A_1,\dots,A_N\subset Q$ such that \[ λ_d(A_n)=\frac1N,\qquad A_n\subset B(x_n,6\sqrt d\,N^{-1/d})\qquad(1\le n\le N), \] and $λ_d\!\bigl(Q\setminus\bigcup_{n=1}^N A_n\bigr)=0$. In other terms, every prefix of the sequence supports an exact transport allocation of Lebesgue mass to its points with uniformly controlled radius $O(N^{-1/d})$. By an elementary partition criterion, this yields \[ W_\infty\!\left(\frac1N\sum_{n=1}^Nδ_{x_n},\,λ_d\right)\le 6\sqrt d\,N^{-1/d} \qquad(N\in\mathbb{N}). \] The bound holds for every $1\le p\le\infty$. The exponent $1/d$ is optimal, so it gives the sharp uniform prefix rate on the cube. The result settles Steinerberger's problem for all $d\ge1$ and all $1\le p\le\infty$.
2603.29600Mar 2026
View2603.29145Bourgain's projection theorem over normed division algebras
We give a simple and self-contained proof of an extension of a projection theorem of Bourgain over the reals to division algebras over local fields of zero characteristic.
2603.29145Mar 2026
View2603.28839Meta Algebras and Special Functions: the Racah Case
Finite families of biorthogonal rational functions and orthogonal polynomials of Racah-type are studied within a unified algebraic framework based on the meta Racah algebra and its finite-dimensional representations. These functions are identified as overlap coefficients between eigensolutions of generalized and standard eigenvalue problems posited on the representation space. The approach naturally yields their orthogonality relations and bispectral properties.
2603.28839Mar 2026
View2603.28223Hypercontractivity of Poisson Semigroups with Orthogonal Polynomial Eigenfunctions
For any $1 < p < q < \infty$, we investigate fixed-time hypercontractive bounds from $L^p$ to $L^q$ of Poisson semigroups associated with the Ornstein--Uhlenbeck, Laguerre and Jacobi operators. We prove that, in the Ornstein--Uhlenbeck and Laguerre cases, the Poisson semigroups fail to be $L^p \to L^q$ bounded for any fixed $t > 0$. In contrast, for Jacobi operators with $α, β\ge -1/2$, the associated Poisson semigroups are ultracontractive, namely bounded from $L^1$ to $L^\infty$. More generally, we study Bernstein subordinations of these semigroups and show that fixed-time hypercontractivity is not stable under subordination. The analysis relies on quantitative $L^q$-estimates for the corresponding orthogonal polynomial eigenfunctions, together with a bilinear test with the exponential family.
2603.28223Mar 2026
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Buffon Discrepancy and the Steinhaus Longimeter
Let $Ω\subset \mathbb{R}^2$ be a convex set. We study the problem of distributing a one-dimensional set $S$ with total length $L$ so that for any line $\ell$ in $\mathbb{R}^2$ the number of intersections $\#(\ell \cap S)$ is proportional to the length $\mathcal{H}^1(\ell \cap Ω)$ as much as possible; we use the term Buffon discrepancy for the largest error. A construction of Steinhaus can be generalized to prove the existence of sets with Buffon discrepancy $\lesssim L^{1/3}$. We also show that the unit disk $\mathbb{D}$ admits a set with uniformly bounded Buffon discrepancy as $L \rightarrow \infty$.
2603.27807Mar 2026
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