Papers

Consider the Mathieu operator $L=-d^2/dx^2+2a\cos(2x)$ for fixed $a\in\mathbb{C}\setminus\{0\}$. For large enough $n$, the periodic ($n$ even) or anti-periodic ($n$ odd) problem on $[0,\pi]$ has exactly two algebraically simple eigenvalues $\lambda_n^-,\lambda_n^+$ in the disc $|\lambda-n^2|<1$. We prove that, as $n\to\infty$, $\lambda_n^+-\lambda_n^-=\pm\frac{8(a/4)^n}{[(n-1)!]^2}\exp\left[-\frac{a^2(n+2)}{4(n^2-1)^2}-\frac{15a^4}{128n^7}+O(n^{-8})\right]$, where the sign depends on the labelling. For real $a$ with $\lambda_n^-<\lambda_n^+$, this difference is the length of the $n$th spectral gap. Expanding the exponential determines the relative corrections of orders $n^{-4}$ to $n^{-7}$, beyond the correction $-a^2/(4n^3)$ obtained by Anahtarci and Djakov (2012) with a relative $O(n^{-4})$ remainder. By isospectrality, the same formula holds for the potential $ce^{-2ix}+de^{2ix}$ with $a^2=cd\ne0$.

We obtain the $L^3$ reverse square function estimate for the light cone in four dimensions from the critical local-smoothing and round-cone propagation statements in the September 2026 OpenAI manuscript. We state these inputs, together with the plate maximal theorem of Di Plinio and Parissis, in a preliminaries section. The proof passes from geometric concentration of packet coefficients to regularity of the analyzed wave, and then from free waves to arbitrary functions supported near the cone. As a consequence, we obtain the corresponding $L^3$ estimate for the sphere and elliptic paraboloids in three dimensions.

Let $\Theta\subset\mathbb{R}^n$, $n\geq 1$ be compact with upper Minkowski dimension $\beta$, and let $\gamma(x,t,\theta)$ be a family of curves which is bi-Lipschitz in the first variable and Lipschitz in the last two. We prove the maximal estimate $$\left\|\sup_{(\tau,\theta)\in[0,1)\times\Theta}\left|e^{i\tau\Delta}f(\gamma(\,\cdot\,,\tau,\theta))\right|\right\|_{L^2(B^n_1)}\lesssim_s\|f\|_{H^s(\mathbb{R}^n)},$$ for every $s>\tfrac{n(\beta+1)}{2(n+1)}$, in the range $0\le\beta\le n-1$. In the range $n-1<\beta\leq n$ we prove the threshold $s>\min\big\lbrace\frac\beta 2 + \frac 1{2(n+1)},\frac n2\big\rbrace$. The proof relies on a fractal $L^2$ estimate of Du–Zhang. We also use the temporal localization of Lee, the locally constant property, and a counting argument allowed by the Lipschitz properties of the curve.

We study the decision problem of global asymptotic stability (GAS) of the origin for homogeneous polynomial vector fields with rational coefficients. In the first part of the paper we give, for every fixed dimension $n$ and odd degree $2k+1$, a computable map which sends a homogeneous vector field of degree $2k+1$ on $\mathbb{R}^n$ to a homogeneous cubic vector field on $\mathbb{R}^m$, $m=\binom{n+k-1}{k}$, and preserves both GAS and Lyapunov stability. The map passes to new variables $y_\alpha=x^\alpha$ (one for each monomial $x^\alpha$ of degree $k$), in which the vector field becomes cubic, and adds a cubic penalty term transverse to the set of points of the form $(x^\alpha)_{|\alpha|=k}$; the main point is that this controls every initial state of the new system. As a consequence, the GAS problems for all degrees, for each fixed odd degree $d\ge3$, and for degree three are pairwise many-one equivalent. In the second part we prove that there are a fixed dimension $N_*$ and a fixed odd degree $d_*$ such that the set of homogeneous vector fields on $\mathbb{R}^{N_*}$ of degree $d_*$ whose origin is GAS is many-one complete. The reduction encodes an undecidable family of polynomial equations in integer unknowns into the constant term of a product of Jacobi theta series, realizes these series as outputs of a polynomial differential system with rational coefficients, and turns the existence of an integer solution into the sign of a maximal invariant average, which homogeneous GAS detects. Combining the two parts shows that GAS of rational homogeneous vector fields on $\mathbb{R}^N$ of degree $d$ is many-one complete for every odd $d\ge3$ and every sufficiently large $N$; in particular it is undecidable for cubic vector fields in a fixed dimension. Building on recent work, some analogous results are proven for Lyapunov stability of homogeneous vector fields.

We study fully nonlinear uniformly elliptic equations that are imposed only when the gradients of solutions lie outside a closed ball and a closed purely one-unrectifiable set that can be unbounded. We prove Harnack inequalities and Hölder regularity estimates for viscosity solutions, with constants independent of the exceptional set. The proofs use the geometry of the contact sets in the cusp and barrier arguments to exclude exceptional gradients. We also provide examples and counterexamples showing the role of pure one-unrectifiability. Materials and proofs are presented in detail to ensure the paper is fully self-contained and accessible to a broad audience.

While models of undirected and directed geometry in random i.i.d. disorder with rapidly decaying tails are expected to behave similarly and exhibit features of the Kardar–Parisi–Zhang universality class, the scenario when the background noise is fractal is expected to be significantly different. A canonical example of the latter is Liouville Quantum Gravity, where the planar Gaussian free field forms the fractal background. A toy model capturing some of the essential features is given by Mandelbrot's fractal percolation, a random Cantor set. In the simplest setting, dyadic cubes of different scales are retained independently with probability $p$ (a parameter of the model), and the intersection of all retained cubes forms the set of open sites. Connectivity properties of such sets have been intensely studied. Across [CCD88, Cha95], a surprising result was proven. Namely, while there is a phase transition for connectivity of the fractal percolation set (like in usual bond percolation), directed percolation never occurs. Results of a similar spirit have been obtained in geometric measure theory as well. As a step towards understanding last passage percolation (LPP) driven by the Gaussian free field (initiated in [GGN24]), we study the problem of quantifying LPP in fractal percolation. Developing a multi-scale framework, we show an almost polynomial gap from linear growth of the passage time. The obstruction to high LPP values is the Lipschitz nature of directed paths. Along these lines, [Cha96] showed that any connected subset of fractal percolation must have Hausdorff dimension strictly larger than one. We show that one can construct directed paths passing through open sites in fractal percolation provided they move exponentially fast in the spatial direction—a model we introduce and term as unrectifiable directed percolation.

Due to the availability of powerful analytic techniques, vastly superior lower bounds are known for the measurable chromatic number of Euclidean spaces compared to their ordinary chromatic number. Indeed, even the breakthrough lower bound of 5 for the famous Hadwiger-Nelson problem lagged over 35 years behind that of the measurable setting. This raises the fundamental question of whether the measurable and ordinary chromatic number of Euclidean spaces differ as conjectured by Székely in 1984. Our main result is that $\overline\alpha(\mathbb{R}^4)=m_1(\mathbb{R}^4)$ and $\chi(\mathbb{R}^4)=\chi^{(m)}(\mathbb{R}^4)$, and for $d\ge5$ that \[ \overline\alpha(\mathbb{Q}^d) = \overline\alpha(\mathbb{R}^d)=m_1(\mathbb{R}^d) \qquad\text{and}\qquad \chi(\mathbb{Q}^d) = \chi(\mathbb{R}^d)=\chi^{(m)}(\mathbb{R}^d). \] Our theorem also holds for multiple forbidden distances $D=\{d_1,\ldots,d_t\}$ provided that $d_1^2,\ldots,d_t^2 \in \mathbb{Q}$. As a consequence, we immediately lift numerous measurable chromatic number results into the ordinary setting. We also take the opportunity to further optimize the new bounds. For multiple distances, Erdős asked whether the chromatic number of $\mathbb{R}^d$ with up to $k$ forbidden distances $D$ grows exponentially in $k$. By a theorem of Bukh, we obtain for $d \ge 4$ that \[ \sup_{|D|=k}\chi_D(\mathbb{R}^d) \ge m_1(\mathbb{R}^d)^{-k}. \] Making progress on another problem of Erdős, we prove that \[ (2+o(1))^d \le \chi(\mathbb{R}^d) \le \left(\frac{3\sqrt{3}}{4}+o(1)\right)^d. \] We also vastly improve the lower bounds for $\chi(\mathbb{R}^d)$ for small $d\ge4$. We expect that our techniques could be developed much further. This includes the possibility of extending our main theorem that $\chi(\mathbb{R}^d)=\chi^{(m)}(\mathbb{R}^d)$ for $d\ge 4$ to $d=3$ or possibly even to $d=2$ to tackle the Hadwiger-Nelson problem.

We identify explicit closed-form and variational formulae for interpolating the exact values and derivatives through order $s$ of a $C^{s,1}$ function $f:\mathbb{R}^d\to\mathbb{R}$ at $N$ distinct points in $[0,1]^d$. The reconstruction satisfies bounds on its global $C^{s,1}$ seminorm and its Lipschitz constant on $[0,1]^d$ that are independent of the sample size $N$, the latter being the sharp Whitney condition. For $s\ge2$, the reconstruction is real analytic away from the data points and definable in the o-minimal structure $\mathbb{R}_{\exp}$. Our nonlinear extension formula coincides with the formula of McShane (1934) for $s=0$ and with the formulae of Le Gruyer and Phan (2015) and Azagra, Le Gruyer, and Mudarra (2018) for $s=1$, and is new for $s\ge2$. For fixed $d$ and $s\ge2$, our closed-form formula is computable away from the data points by a circuit using only elementary unary and binary real operations, with $\mathcal{O}(N)$ gates and $\mathcal{O}(\log N)$ depth. This yields $\mathcal{O}(N)$ storage and $\mathcal{O}(\log N)$ parallel evaluation time. From the supplied coefficients and weights, the circuit can be compiled with $\mathcal{O}(N)$ one-time work in the exact-real word-RAM model. Thus, in this supplied-data setting, our nonlinear construction improves by a logarithmic factor on the $\mathcal{O}(N\log N)$ initialization bound of Fefferman and Klartag (2009), while retaining linear storage and attaining logarithmic query time through parallel evaluation.

The second part of Hilbert's 16th problem is one of the most fundamental open problems in the qualitative theory of ordinary differential equations and dynamical systems. A central unresolved question is whether the Hilbert number $H(d)$ is finite for arbitrary polynomial degree $d$. (A very recent manuscript from OpenAI asserts the finiteness of $H(d)$; see the remark at the end of this paper.) In the absence of a general upper bound theory, most existing work constructs explicit perturbed polynomial systems to obtain improved lower bounds for $H(d)$, while rigorous upper bound estimates remain largely unavailable. This paper focuses on improving the quantitative lower bound for polynomial systems through a systematic Hierarchical Melnikov Realization framework. We construct perturbed dynamics based on an anisotropic Chebyshev Hamiltonian, where hierarchically selected perturbation coefficients generate simple zeros of the first-order Melnikov function simultaneously across multiple disjoint period annuli. Two distinct 3-adic filtrations supply the necessary logarithmic correction factors, and a Borel–Gauss analysis establishes the required local rank condition under sufficiently strong anisotropy. As the main result of this work, we prove that $H(d)=\Omega\bigl(d^2\ln^2 d\bigr)$, which improves the previously known lower bound by a factor of $\ln d$. The full construction of limit cycles and the resulting dynamical bounds are formally certified using Lean 4 formal verification.

For a non-empty bounded set $E\subset\R^d$, the upper and lower box dimensions of its orthogonal projections are constant onto almost all $m$-dimensional subspaces. These almost-sure values are called the upper and lower $m$-box dimension profiles of $E$, respectively. We study their relationship with intermediate dimensions, which interpolate between Hausdorff and box dimensions by restricting the relative sizes of covering sets. We establish several equivalent characterizations of both quantities and obtain upper and lower bounds for box dimension profiles in terms of intermediate dimensions, with the lower bounds also involving the upper Assouad spectrum. We further derive explicit formulas for the box dimension profiles in terms of the box dimension and the initial growth rate of the intermediate dimensions, under a suitable condition on the quasi-Assouad dimension.

This paper concerns the planar Liénard system \(\dot x=y-F(x),\ \dot y=-x\), where \(F(x)\) is a piecewise linear function with exactly \(m\) jump points and no fold points. Tonnelier [SIAM J. Appl. Math. 63 (2002)] conjectured that the maximum number of limit cycles of the system is \(2m\). The conjecture was confirmed for \(m=1\) in [J. Lond. Math. Soc. 113 (2026)], and a lower bound of \(2m\) for arbitrary \(m\) was established by Chen et al. [arXiv:2608.19542]. In this paper, we construct systems with at least \(4m-2\) hyperbolic crossing limit cycles for every positive integer \(m\), thereby disproving Tonnelier's conjecture for \(m\geq2\). We also establish the uniform upper bound \(2^{224(m+1)^2}\) for the total number of crossing, grazing, sliding, and composite limit cycles.

We construct nets of curves in a rectangle $R$ with special properties. The net is formed by two families of curves, "horizontal" and "vertical", each family foliates $R$, and every horizontal curve meets every vertical curve orthogonally. Moreover, every horizontal curve started at the left edge of $R$ arrives to the right edge of $R$ at the same height, and every vertical curve started at the bottom edge of $R$ arrives to the top edge of $R$ at a prescribed in advance longitude. The latter longitude function as a map from bottom to top can be chosen in advance as any small perturbation of the identity. Our motivation for this was to design some elliptic divergence form operators, for instance on one of the domains bounded by a Reifenberg flat curve in the plane, with the scalar coefficient and the corresponding elliptic measure being absolutely continuous with respect to the natural measure on the boundary (an object of a separate paper). The construction of our net of curves in a rectangle can also be interpreted as control problem, has an interest on its own, and, likely, has other applications.

We resolve Erdős Problem #522, posed in 1961, by proving that for random Littlewood polynomials the proportion of zeros in the closed unit disk converges almost surely to $1/2$. More generally, for partial sums of random power series with i.i.d. coefficients that are bounded, nondegenerate, and centrally symmetric, real Gaussian, or circular complex Gaussian, we prove that the proportion of zeros in the disk of radius $1+x/n$ converges almost surely to $\frac12(1+\coth x-1/x)$, uniformly in $x\in\mathbb{R}$. Thus almost all zeros lie at distance $O(1/n)$ from the unit circle. For random Littlewood polynomials, even with arbitrary dependence between degrees, the convergence at the unit circle holds at the almost-sure rate $O(n^{-1/4}\sqrt{\log n})$. We establish a law-of-the-iterated-logarithm criterion for dependent sequences and, for real Gaussian coefficients, obtain the sharp law of the iterated logarithm over all integer degrees, variance and cross-degree covariance asymptotics, quantitative central limit theorems, joint Gaussian limits, and increment bounds. For symmetric coefficient laws with bounded density and all moments finite, the number of zeros in the closed unit disk has variance $(c_G+\kappa_4/12)n+O_\xi(n^{399/400})$, and for random signs the constant is $c_G-1/6$. Thus the radial law is universal, while the fluctuations depend on the coefficient law through its fourth cumulant $\kappa_4$. All our main results are formally verified in Lean 4, and our proof of Erdős Problem #522 has been accepted as a solution to its formal statement in Google DeepMind's Formal Conjectures project.

We establish an exact Fourier restriction criterion for a class of weighted Moran measures. Assuming uniformly bounded one-level frame bounds, we prove that, for every \(s>2\), the measure itself already detects the full extension estimate: \(L^2(\mu)\to L^s(\mathbb R)\) extension holds if and only if \(\widehat\mu\in L^s(\mathbb R)\), equivalently, if and only if the one-level Fourier moments \(G_{s,n}\) are summable, where $G_{s,n}$ is the $s$th moment of the nonzero Fourier coefficients of the $n$th digit law. The sufficient direction is obtained from a finite-frame stability estimate and its multiscale iteration, while necessity follows from disjoint resonant frequency grids. This gives, in particular, an exact formula for the critical restriction exponent and a precise criterion for endpoint attainment. The criterion reveals that the optimal restriction range is not determined by standard Fourier or dimensional data. We construct singular continuous spectral Moran measures with the same scales, digit cardinalities, exponential spectrum, and maximal nonzero Fourier coefficients. They also satisfy the same global Fourier decay bound, which is sharp along the same sequence of resonant frequencies, yet their critical restriction exponents are different. More generally, after fixing the summability rate of the maximal Fourier coefficients, we determine the optimal interval of possible restriction exponents and show that, at every interior critical exponent, the same fixed data allow both attainment and failure of the critical endpoint. These measures may simultaneously have $ \dim_{\mathrm F}^{\theta}\mu=\theta$ and $ D_q(\mu)=1,$for all admissible $\theta$ and $q$, and pointwise local dimension one everywhere on their supports.

We obtain the necessary and sufficient conditions on input functions for a family of positive exponent power weight $L^1$ Stein-Weiss inequalities. Earlier work by De Nápoli and Picon showed that a certain cocanceling condition was necessary and sufficient in $L^1$, but it was unknown if this condition remained necessary in the positive exponent power weight setting. This present article identifies weaker necessary conditions for positive exponent power weights, some of which were shown to be sufficient for positive, non-integer exponent power weights in the author's earlier work.

We find the exact rate of equidistribution of several arithmetic point configurations in Wasserstein metric and $L^2$ discrepancy with respect to balls. Kronecker sequences in the torus and the Fibonacci lattice on the Euclidean unit sphere are shown to be more evenly distributed in Wasserstein metric than random points. Modular hyperbolas and modular parabolas in the torus behave like random points, although the constants in the asymptotics of the $L^2$ discrepancy with respect to balls differ from the random case. We also prove a sharp inequality between the quadratic Wasserstein metric and the $L^2$ discrepancy with respect to balls for an arbitrary probability measure on the torus or the sphere. In particular, we show that any point set that is optimally close to the uniform measure in $L^2$ discrepancy with respect to balls is also optimal in the quadratic Wasserstein metric.

We show that $m$-linear homogeneous rough singular integral operators $\LL_{\Omega}$ associated with $L^\infty$ functions $\Omega$ on $\mathbb{S}^{mn-1}$ with mean value zero may not be bounded from $L^1(\mathbb R^n)\times\cdots\times L^1(\mathbb R^n)$ to $L^{\frac 1 m,\nf}(\mathbb R^n)$ except possibly for the case $n=1$ and $m=2$.

In this paper we propose and study what we call the Erdős-Falconer distance problem in the integer lattice. One motivation is Magyar's discrete analog of the result of Furstenberg, Katznelson, and Weiss, who showed that all arbitrarily large distances had to appear for a set of positive upper Banach density in $\mathbb{R}^d$, $d\geq 2$. Another motivation is the Falconer distance problem, which can be viewed as a refinement of a result of Steinhaus, dropping assumptions from positive Lebesgue measure to merely asking about the dimension of sets. In this context we thus want to lower the assumption of positive upper Banach density and yet obtain statements that guarantee that we have lots of distinct distances. All of these questions are also related to the Erdős distinct distance problem. Our naming is as for the finite field variant of the problem, acknowledging that parts of the problem share similarities with the Erdős distinct distance problem while others share similarities with the Falconer distance problem. The main tool we use are $\ell^p$-improving inequalities for discrete spherical averages, which to the best of our knowledge is the first application of such bounds. One can use results on the Erdős distinct distance problem to make progress on our conjecture but surprisingly, even if using the strongest available bounds in high dimensions using the mechanism of Solymosi and Vu, then in certain regimes our methods do better. While a resolution of the Erdős distinct distance problem would imply our unpinned conjecture then no results on either the Erdős or Falconer distance problem seem to imply any results for our pinned variant. We view this problem as a model problem that might ultimately contribute back to the original pinned distance problems.

We characterize Fourier quasicrystal measures with positive integer masses as real sections of effective Lee-Yang cycles, with weighted local intersection degrees. The section matrix has positive maximal minors, and the mass at each root is its weighted local intersection degree. For Delone support the reduced support of the cycle can be chosen strict. The support of every integer-mass Fourier quasicrystal measure is the common complex zero set of finitely many real-valued trigonometric polynomials. For unit masses the representing cycle can be chosen reduced, with local degree one at every physical root. These results answer Questions (1), (2), and (4) of Alon, Kummer, Kurasov, and Vinzant. The proof recovers a finite-dimensional torus from the spectrum, proves that the resulting analytic set is algebraic, and reconstructs its complex branches and their integer coefficients from their moments. A support-height theorem for holonomic sequences gives the required linear spectral bound. Singular Lee-Yang sections are Fourier quasicrystal measures with their natural local intersection degrees; discarding those degrees can destroy the Fourier quasicrystal property.

We study the $(n-2)$-plane Kakeya problem in $\mathbb{R}^n$ ($n\geq4$) subject to convex Wolff axioms. In other words, we study finite families of $\delta$-neighborhoods of $(n-2)$-plane segments ($\delta$-flakes) in $\mathbb{R}^n$, with the property that not too many of them can be contained inside a common convex set. Our main result is that the union of $\delta$-flakes from such a set must have almost maximal volume. This can be viewed as a higher-dimensional generalization of Wang and Zahl's work for $\delta$-tubes in $\mathbb{R}^3$. The latter implies the Kakeya set conjecture in $\mathbb{R}^3$. As a consequence, we resolve Rogers' $2$-plane Kakeya problem in $\mathbb{R}^4$ subject to certain curvature conditions on two-dimensional families of direction subspaces in the Grassmannian $\mathrm{Gr}(4,2)$.

For each real $b$, exact upper and lower bounds on the probability $\mathsf P(X\ge b)$ over all random variables $X$ with log-concave p.d.f.'s such that $\mathsf E X=0$ and $\mathsf E X^2=1$ are obtained, as well as the best constant factor $C$ in the inequality $\mathsf P(X\ge b)\le C e^{-b}$ for all real $b\ge0$. Exact upper and lower bounds on the corresponding exponential moments (Proposition 9.4) and explicit exponentially decreasing upper bounds on the mentioned p.d.f.'s (Propositions 9.2 and 9.3) are given as well. Some general results concerning log-concave p.d.f.'s are also obtained.

We study the exact $L^p$ norms for a number of maximal operators, which are prominent in harmonic analysis, semigroup theory, and probability. We prove that a strongly continuous positive symmetric sub-Markovian semigroup $(T_t)_{t\ge0}$ on a nonatomic $\sigma$-finite measure space $(X,\mathcal F,\mu)$ has maximal $L^p$ norm $p'=p/(p-1)$ for every $1<p<\infty$ whenever $f\mapsto T_{t_0}(\mathbf 1_Ef)$ is compact on $L^2(\mu)$ for some $t_0>0$ and some measurable set $E$ of positive measure. The lower bound follows by approximating finite dyadic conditional expectations at suitable semigroup times; the upper bound is Stein's maximal inequality in its sub-Markovian form. Important examples of such maximal operators we discuss include maximal functions of various semigroups (heat, Poisson, Ornstein–Uhlenbeck, Schrödinger, Dunkl, Laguerre, Jacobi) as well as the centered Hardy-Littlewood maximal operator on $\mathbb{R}.$ In particular, the heat and Poisson maximal operators on every complete connected Riemannian manifold of positive dimension without boundary have exact $L^p$ norm $p'$, with no curvature or stochastic completeness assumptions. Combining our approach with previous work of the first and last authors we also prove that $p'$ is the limit, as $d\to \infty,$ of the $L^p$ norms $B(p,d)$ of the centered Hardy-Littlewood maximal operators on $\mathbb{R}^d.$ On the other hand, we show that the norm of this maximal operator is strictly larger than $p',$ in dimensions $d\ge2.$

We consider the Dirichlet problem for the Lorentz–Minkowski mean curvature operator in a radial setting, $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right) = \lambda u + \mu g(u) \quad \text{in } \mathcal B_R, \qquad u = 0 \quad \text{on } \partial\mathcal B_R, $$ on a ball of radius $R$ centered at the origin $\mathcal B_R \subset \mathbb{R}^N$, where $\lambda, \mu \geq 0$ are parameters, and the nonlinearity $g$ satisfies suitable growth assumptions at zero. We first prove an a priori dichotomy for the asymptotic behaviour, as $\mu \to +\infty$, of solutions with a prescribed number of zeros: either they converge uniformly to $0$, or they converge to a piecewise affine limit function $u_\infty$, with $|u_\infty'|=1$ a.e., having the same nodal properties. We then show that, when the number of zeros is appropriately prescribed, both alternatives are actually realized. More precisely, building on the multiplicity result in [A. Boscaggin, F. Colasuonno, and R. Ziegele, Positive and nodal solutions for the Minkowski mean curvature equation: multiplicity and asymptotics, arXiv:2607.15956 (2026)], we refine the shooting construction therein to obtain a family of small solutions converging to zero and, through a backward shooting argument, a family of large solutions converging to a saturated piecewise affine profile.

For every $1<p<\infty$, the Hilbert transform on $L^p(\mathbb{R};\mathcal{S}_p)$ and the ordered Hilbert Schur multiplier on $\mathcal{S}_p$ have norm $C_p=\cot(\pi/(2p^*))$, where $p^*=\max\{p,p/(p-1)\}$. The scalar Hilbert transform and the Schur multiplier have the same canonical completely bounded norm. This proves the Gohberg-Krupnik sharp Volterra conjecture. A simultaneous finite-dimensional realization also resolves the Riesz-Titchmarsh conjecture and Laeng's Conjecture 5.7 for all shifts. The norm identities extend to finite families of real Hilbert-transform pencils with $\ell^p$ direct-sum ranges. The proof uses a spectral-defect identity with two nonnegative remainders, based on Heinävaara's tracial joint spectral measure. It classifies trace inequalities for finite signed real ridge sums and identifies the least homogeneous superharmonic majorant as the pointwise least additive eigenvalue correction. The associated angular cone is characterized by nonnegative potentials and realized by weak-* limits of weighted angular measures of finite nilpotent matrices.

This paper concerns the planar Liénard system \(\dot x=y-F(x),\ \dot y=-x\), where \(F(x)\) is a continuous piecewise linear function with exactly \(n\) fold points. Tonnelier [SIAM J. Appl. Math. 63 (2002)] conjectured that the maximum number of limit cycles of the system is \(n\) when \(F\) has exactly \(n\) fold points. The conjecture was confirmed for \(n=1\) and \(n=2\) in [J. Nonlinear Sci. 25 (2015)] and [J. Lond. Math. Soc. 113 (2026)], respectively, whereas the case \(n\ge3\) remained open. In this paper, we show that the maximal number of limit cycles is at least \(|3n-4|\) for \(n\in\mathbb N^+\), thereby disproving Tonnelier's conjecture for \(n\ge3\). The proof reveals a unified multiscale perturbation mechanism underlying the creation and coexistence of multiple limit cycles. Moreover, we prove that the number of limit cycles admits the finite upper bound \(2^{28(n+1)^2}\).

In the framework of multiple orthogonal polynomials (MOPs) and their extension to the multivariate setting, a unified mechanism for generating structured families of multiple orthogonal polynomials across diverse multidimensional domains is still lacking. In this work, we bridge this gap by developing an extended Koornwinder-type methodology for constructing multivariate multiple orthogonal polynomials. We introduce two distinct coupling schemes: combining a univariate MOPs family with a standard orthogonal family, which enables the definition of both Type I and Type II bivariate systems along with their dual biorthogonality relations, and coupling two univariate MOPs families to form Type II multivariate systems. Using this general framework, we provide the first explicit formulations of multiple orthogonal polynomials on a variety of bivariate regions, including bounded domains such as a parabolic domain and the square $[0,1]^2$, non-standard unbounded geometries like the positive quadrant $(\mathbb{R}_0^+)^2$ and the wedge $\mathbb{V}^2$, as well as new constructions on the triangle $T$. Furthermore, explicit multiple orthogonal systems on the $d$-dimensional simplex $T^d$ and on the $d$-dimensional cone $\mathbb{V}^d$ are given.

Mourad E.H. Ismail and Kenneth S. Miller in 1982 conjectured that for $\nu>\mu\geq0$ and $b>a>0$ the function $$x\mapsto G(x;\mu,\nu)=\left(\frac ba\right)^{\mu-\nu}\frac{I_\mu(a\sqrt{x})I_\nu(b\sqrt{x})}{I_\mu(b\sqrt{x})I_\nu(a\sqrt{x})},$$ where $I_\nu$ stands for the modified Bessel function of the first kind, is the Laplace transform of an infinitely divisible probability distribution. Their conjecture is equivalent to the monotonicity property of some exponential sums over the zeros $j_{\nu,n}$ with respect to the order, where $j_{\nu,n}$ denotes the $n$th positive zero of the Bessel function of the first kind of order $\nu$. In this paper our aim is to reduce the problem further to the monotonicity, with respect to $\nu$, of \[Q_\nu(t)=\sum_{n\geq1}j_{\nu,n}^2e^{-t j_{\nu,n}^2},\qquad t>0,\] and to prove this monotonicity. In this way we show that indeed the above function $x\mapsto G(x;\mu,\nu)$ is the Laplace transform of an infinitely divisible probability distribution, and the above conjecture is true. The key ingredient in our proof is an integral representation for the derivative of a quotient of modified Bessel functions of the first kind with respect to the order, combined with Bernstein's theorem and a probability tail argument.

We develop a sparse domination theory that captures both local polynomial approximation and cancellation of the input below the scale of each sparse cube. Its coefficients are percentiles of localized smooth maximal functions of the input minus its local polynomial approximation, combining features of mean-oscillation sparse bounds and of the cancellative sparse bounds of Conde-Alonso, Lorist and Rey. Using wavelet representations, we characterize the Calderón-Zygmund operators whose derivatives admit these bounds by explicit polynomial testing conditions. In the homogeneous setting, the same conditions characterize weighted Hardy-Sobolev inequalities with linear dependence on the $A_\infty$ characteristic. This quantitative dependence detects the transition from BMO to $L^\infty$ testing conditions that Lerner found for pointwise bounds by Riesz potentials, a distinction that qualitative weighted estimates cannot recover. In the inhomogeneous setting, we prove corresponding weighted inequalities and show that they are strictly weaker than pointwise sparse domination from the second derivative order onward. Consequences include Hardy-Sobolev estimates below the Banach range, sharp weighted Sobolev inequalities, and pointwise bounds by Riesz potentials.

The full shadow of a complex polynomial $P$ of degree $d\ge2$ is the closure of all zeros of $(P^n)^{(m)}$, $n\ge1$, $0\le m<dn$. We prove that it consists of a compact connected limiting core containing all roots and critical points of $P$, together with isolated transient zeros, each occurring at only finitely many powers. The core is the Hausdorff limit of the zero sets pooled over all derivative orders as $n\to\infty$ and the support of their limiting normalized root-counting measure. For $m/n\to\alpha\in(0,d)$, we establish weak convergence of normalized root-counting measures and Hausdorff convergence of zero sets to the limiting supports, both stable under coefficient perturbations. We exhibit quartics with infinitely many transients. A bounded operator constructed from polynomial differentiators realizes the full shadow, core, and transients as its spectrum, Fredholm essential spectrum, and discrete spectrum, respectively.

Let $K=K(n,m,\Gamma)$ be a Bedford-McMullen carpet with $n>m\geq2$ and $1<\#\Gamma<nm$, and let $\mu$ be a self-affine measure on $K$ with strictly positive weights. We prove that if a similitude $f$ satisfies $\mu(K\cap f(K))>0$, then its orthogonal part is diagonal or anti-diagonal. This holds without any arithmetic restrictions on the defining bases. For isometries, we further prove that if $\mu(K\cap f(K))>0$ then $f(K)$ and $K$ differ only by a translation, thereby establishing a strengthened form of the 2010 conjecture of Elekes, Keleti and Máthé in this planar self-affine context.