Papers

A blob grows on the plane (as in films called "The Blob" of 1958 and 1988). Suppose that it grows in all directions at unit rate, and can be stopped only by a certain kind of fence that can be manufactured at rate $\lambda$. What is the critical value $\lambda_C$ for this rate, above which the blob can be surrounded and contained, and below which we are all doomed? This problem, introduced by Bressan in 2007 and independently by Barghi and Winkler in 2013, was in both cases motivated by consideration of strategies in fighting forest fires. In particular, does it pay to place "gambit" barriers which will subsequently be overwhelmed by the fire, in order to slow it down enabling later containment? A non-gambit strategy, which will be described, succeeds when $\lambda > 2$ and is widely believed to be optimal; but up until now no one has succeeded in obtaining a lower bound greater than 1 for $\lambda_C$. We introduce a novel weakened version of the blob which is used to raise the lower bound to 1.5.

We give a self-contained proof, in the bounded variation setting, of the Hambly–Lyons uniqueness theorem, which states that (total) signature identifies the path up to tree-like equivalences. The argument is organized around two key geometric observations. First, tree-like paths have trivial signature because factorization over a loop in a tree $\tau:[0,1]\to T$ is preserved under signature lifts, which follows from an elementary property of planar curves. Second, a path with trivial total signature contains a nontrivial subpath with trivial total signature (the sub-interval lemma). This is proven by applying a winding-number argument to a two-dimensional projection of the signature lift. Collapsing all trivial-signature sub-intervals then defines a compact metric tree $T$ through which the original path factors by virtue of the sub-interval lemma.

We introduce a new method for obtaining sharp $L^p$ estimates for martingales under predictable analogues of differential subordination. By allowing suitable comparisons between continuous and jump variation adapted to the relevant range of $p$, we retain sharp constants in settings where pathwise differential subordination fails. The key idea is to study together the continuous and jump contributions arising from the appropriate Bellman functions. Although these contributions need not be nonpositive separately, we quantify their defects and show that they compensate at the predictable level. This compensation mechanism is inspired by the author's earlier work. As an application, we substantially improve the explicit dimension-free bounds for the Riesz vectors on the Hamming cube and on $\mathbb{Z}^n$.

Let $\Omega\subset\mathbb Z^d$ be finite and let $S\subset\mathbb T^d$ be measurable. We study the discrete Fourier concentration operator \[ T_{\Omega,S}=P_\Omega B_SP_\Omega, \] which describes simultaneous localization to a finite set of spatial indices and a prescribed spectral region. For independent exponents $\gamma,\eta\in(0,1]$, we introduce translation seminorms for $\Omega$ and $S$ and use them to bound the trace defect \[ \operatorname{tr}(T_{\Omega,S}-T_{\Omega,S}^2). \] The proof combines an exact trace-defect identity with dyadic Fourier estimates, leading to three regimes: when $\gamma=\eta$, the contributions from the dyadic scales are of the same order, and their sum produces a logarithmic factor, while for $\gamma\ne\eta$ the sum is controlled by one end of the scale range. For discretizations $$ \Omega_R=(RF)\cap\mathbb Z^d $$ of a bounded measurable set $F\subset\mathbb R^d$ satisfying an upper Minkowski-neighborhood estimate with exponent $\gamma$, we obtain trace-defect bounds of order $R^{d-\gamma}\log R$ in the critical case and $R^{d-\min\{\gamma,\eta\}}$ off the critical line. These estimates yield quantitative eigenvalue-counting and plunge-region bounds and recover the discrete Landau asymptotic. For a box model at $\gamma=\eta=1$, the logarithmic factor is sharp.

We develop a martingale-transform framework for Dunkl harmonic analysis and apply it to prove $L^p$ estimates for the Dunkl Riesz transforms using the martingale decomposition of the Dunkl process. A fundamental difference from the classical Brownian setting is that the martingale representing the Dunkl Riesz transform of a function is not, in general, differentially subordinated to the Poisson martingale of the function itself. Our main argument bypasses this obstruction by applying Burkholder's Bellman function directly. The possible positive defect of the continuous Itô drift is compensated by the negative contribution produced by the reflection jumps. This yields $L^p$ estimates for single Dunkl Riesz transforms for $1<p<\infty$, and vector-valued estimates for $p\geq2$ with a spectral dependence on the root system. For $G$-invariant functions, differential subordination can be recovered and the resulting estimates are independent of the root system.

We investigate the maximization problem for the family of mixed-norm Strichartz inequalities for the Schrödinger equation, $\|e^{-it\Delta/2}f\|_{L_t^qL_{\boldsymbol{x}}^r(\mathbb{R}^{1+d})}\le C_{q,r}\lVert f\rVert_{L^2(\mathbb{R}^d)}$, with $2/q+d/r=d/2$, $q,r\geq 2$, and thus $r\leq 2d/(d-2)$ if $d\geq 3$. We show that, in low dimensions $1\leq d\leq 5$, the thresholds $\rho_1=10$, $\rho_2=6$, $\rho_3=4\sqrt{7}-6\approx 4.583$, $\rho_4=2\sqrt{15}-4\approx 3.746$, and $\rho_5=10/3\approx 3.333$ are such that gaussians are stable local maximizers for $2<r<\rho_d$, and fail to be local maximizers for $\rho_d<r\leq 2d/(d-2)$ (with the conventions there is no upper bound on $r$ when $d\in\{1,2\}$ and that $r=\infty$ is excluded when $d=2$). In the cases $(q,r,d)\in\{(6,6,1),(8,4,1),(4,4,2)\}$, we establish global stability inequalities with effective stability constants. Both proofs hinge on spectral gaps which we compute exactly.

We investigate the class of Liouville regular Jordan curves, which was introduced in our previous paper under the phrase "Jordan curves that admit continuous Legendrian lifts". We prove that a Jordan curve satisfying certain variation condition in a neighborhood of each point is in this class. We also prove that the boundary of a Jordan domain is Liouville regular if the images of the radial tails under a conformal mapping have uniformly vanishing length. Together with our previous result, these criteria imply the existence of an inscribed rectangle in every prescribed similarity class.

We provide a shorter and more direct proof of $L^p$ estimates for maximal Riesz transforms (of an arbitrary order) in terms of the corresponding Riesz transforms, with a constant independent of the dimension of the Euclidean space $\mathbb R^d$. This result was originally proved by Mateu, Orobitg, Pérez and Verdera with a constant depending on the dimension, and improved to a dimension-free inequality by Kucharski, Wróbel and Zienkiewicz.

We study when the image of a measure under a random Hölder map attains full packing dimension. Our main tool is a family of packing intermediate dimension profiles $\dim_{P,\theta}^{s}\mu$, indexed by $\theta\in(0,1]$ and $s>0$, which refine the packing dimension profiles of Falconer and Howroyd and reduce to them at $\theta=1$. For a large family of random $\alpha$-Hölder maps $f_\omega:\mathbb{R}^n\to\mathbb{R}^m$, which includes index-$\alpha$ fractional Brownian motion as a particular case, we prove that for every compactly supported Borel probability measure $\mu$ on $\mathbb{R}^n$, $$\dim_P \mu_{f_\omega}=m \quad\text{almost surely} \quad\Longleftrightarrow\quad \alpha m\leq \lim_{\theta\to 0}\dim_{P,\theta}^n \mu.$$ We further study the profiles $\dim_P^s\mu$ themselves, obtaining a quantitative lower bound and a Marstrand-type identity for their limiting behavior as $\theta\to 0$. Finally, we obtain the analogous characterization for analytic sets: $$\dim_P f_\omega(E)=m \quad\text{almost surely} \quad\Longleftrightarrow\quad \alpha m\leq \lim_{\theta\to 0}\dim_{P,\theta}^n E,$$ where $\dim_{P,\theta}^n E=\sup\{\dim_{P,\theta}^n \mu :\, \mu\in \mathcal{M}_c^+(E) \}$.

We investigate sharp inequalities for ratios of Riesz capacities with negative exponents by combining computational experiments with rigorous analysis. For finite subsets of the line, we prove positivity of equilibrium masses when $-1<p<0$, enabling numerical tests of conjectured extremal ratios. In the plane, comparisons of the disk with regular polygon vertex sets reveal a cascade of transitions among the tested competitors and suggest a precise conjecture for the equilibrium measure of odd polygons, for which we give a partial proof. Numerical intersections of equality curves show that the regions where these sets outperform the disk are not simply nested. Similar numerical intersections occur in three dimensions between the regular-simplex equality curve and those of explicit five-point and six-point configurations. Motivated by the dimensional dependence of these comparisons, we prove that for each fixed $p<-2<q<0$, the regular simplex has a larger capacity ratio than the ball in all sufficiently large dimensions. Accompanying Python and Mathematica code supports reproduction and further testing of the conjectures.

For a twice differentiable function $f:\left( a,b\right) \rightarrow \mathbb{R}$ define $v\left( f\right) =f^{\prime}f^{\prime}-f^{\prime\prime }f.$ It is well known that the positivity of $v\left( f\right) $ implies that the function $\left\vert f\right\vert $ is strictly log-concave on each subinterval which does not contain zeros of $f.$ In this paper we provide criteria for the positivity of $v\left( F\right) $ for the composite Bessel function $F\left( x\right) =J_{\alpha,\beta,\gamma,\nu}\left( x\right) :=x^{\alpha}J_{\nu}\left( \beta x^{\gamma}\right) $ for positive numbers $\beta$ and $\gamma$ and real numbers $\alpha$ and $\nu.$

We prove that the full discrete Hardy-Littlewood maximal operator associated with Euclidean balls satisfies dimension-free bounds on $\ell^p(\mathbb Z^d)$ for every $1<p<\infty$. We also establish analogous dimension-free bounds for the full discrete spherical maximal operator when $d\geq 5$ and $2\leq p<\infty$. The main new idea is to approximate the relevant Fourier multipliers by finite linear combinations of normalized discrete Gaussian multipliers and their translates. We obtain these approximations uniformly in frequency and with uniformly bounded coefficients through a refined saddle-point analysis. The ball result resolves a question of E.M. Stein, while the spherical result gives, in the range $p\geq 2$, a dimension-free strengthening of the theorem of Magyar, Stein, and Wainger.

We prove that the intrinsic jump set of every locally integrable function on an equiregular Carnot-Carathéodory space is intrinsically countably rectifiable. A key new ingredient is a local scalar Whitney extension theorem for intrinsic $C^1$ functions.

We characterize equality in the finite free Stam and entropy-power inequalities, proving that Hermite polynomials are the unique extremizers among simple real-rooted inputs, up to independent translations and scalings. The proof turns this classification into a rigidity problem for projective plane curves. Using hyperbolicity and the Helton-Vinnikov theorem, we express the Jacobian defect as an off-diagonal squared norm in a definite symmetric pencil. Together with score transport, this yields a matrix proof of Stam for all real-rooted inputs. For each simple base pair, the directions in which the defect vanishes are the independent translations and common dilation, forming a three-dimensional subspace in every degree whose curves split into $n$ projective lines. At a collision, the leading configurations are again finite free convolutions of normalized derivatives of the velocity polynomials of the colliding input clusters. Combined with incidence counting, this local formula shows that at most one real fiber is singular, every real singularity is an ordinary totally real multiple point, and the ordered collision multiplicities determine the real normalization covering. For $n\geq3$, every non-split curve has at least $2n-2$ non-real projective discriminant zeros, counted with multiplicity, with equality attained by irreducible curves of geometric genus zero through every simple input pair. The leading Fisher-information coefficient is determined by the colliding tangent configurations, while the finite entropy term retains the gaps between clusters. For $n\geq3$, maximal logarithmic entropy divergence along the optimally weighted score direction is equivalent to Stam equality.

We prove the strict positivity conjectured by Chenevier for the critical vectors in the unconditional part of his automorphic Hermite–Minkowski theorem. The proof establishes strict negativity of all Verblunsky coefficients of a circle measure associated with the weight $(\arcsin x)/x$ on $(-1,1)$. A positive-kernel formula and the classical Schur algorithm give these signs, and a para-orthogonal transformation yields positivity in every degree. We also compute the positive density representing the negative of the Schur function as a Hausdorff moment generating function. After rescaling their indices to $[0,1]$, the normalized critical vectors converge weakly to the arcsine law, while Chenevier's critical scale is asymptotic to $8\pi/n$. At the critical boundary, a single nonzero effective integral vector is negative for every admissible test function exactly in odd degree and in degree zero. Finally, we prove an exact first-variation formula for exponential perturbations of the Legendre measure and derive its asymptotics for endpoint cusps. For the perturbation leading to Chenevier's weight, the derivative at the Legendre measure has a $(\log n)/(\pi^2n^2)$ term and an explicit constant at order $n^{-2}$. The corresponding nonlinear asymptotic remains conjectural.

For $-1<\alpha\leq1/2$, let $J_\alpha$ be the Bessel function of the first kind and let $j_{\alpha,2}$ be its second positive zero. Define $\beta(\alpha)<\alpha+1$ by $$ \int_0^{j_{\alpha,2}}u^{-\beta(\alpha)}J_\alpha(u)\,du=0. $$ We prove that $\beta''(\alpha)>0$ for $-1<\alpha\leq1/2$, including the left second derivative at $\alpha=1/2$. The continuous extension $\beta(-1)=0$ is strictly convex on $[-1,1/2]$, strengthening Askey's 1993 convexity conjecture. The analytic argument uses a positive expansion of a primitive and a vanishing weighted sum. Increasing ratios of consecutive weights give a negative covariance term, while one comparison controls slope and curvature. The remaining algebraic step proves eight rational inequalities by finite exact polynomial calculations, reproduced in an appendix.

We establish a Loomis-Whitney inequality for the Reiter-Heisenberg groups $\mathbb{G}_{qp}$, a family of step-two Carnot groups that includes the Heisenberg groups when $q=1$. The proof is based on the duality between Brascamp-Lieb inequalities and entropy subadditivity: we first derive the result for $\mathbb{G}_{q1}$ from the known inequality on the first Heisenberg group, using conditional entropy and the invariance of differential entropy under volume-preserving diffeomorphisms; then we pass from $\mathbb{G}_{q1}$ to $\mathbb{G}_{qp}$ via a stability principle for Loomis-Whitney inequalities under finite central sums, which generalizes the argument in (Zhang, 2024 arXiv:2402.02749v2). As consequences, we obtain the associated geometric projection inequality, a Gagliardo-Nirenberg-Sobolev inequality, and an isoperimetric inequality.

Let $M_t$ denote the normalized average over the lattice points in the Euclidean ball of radius $t$ in $\mathbb{Z}^d$. We prove that the full maximal operator $f\mapsto\sup_{t\geq0}\lvert M_t f\rvert$ is bounded on $\ell^p(\mathbb{Z}^d)$, for every $1<p\leq\infty$, with a constant independent of the dimension. In particular, this resolves a question of E.M. Stein from the mid 1990s. The principal ingredient in our proof is that, when $t\lesssim d$ with $t$ sufficiently large, the associated multiplier $\mathfrak{m}_{\sqrt{\lfloor t^2\rfloor}}(\xi)$ admits an asymptotic expansion of arbitrary prescribed order, uniform in $\xi$, whose resulting maximal operators can be controlled by the discrete normalized Gaussian maximal function studied by Mirek–Szarek–Wróbel \cite{MSW25}.

In this paper we investigate the optimal dimension-dependent estimates of the second-order discrete Riesz transforms \[ R_{\mathrm{dis}}^{(jk)}f(n) = c_d\sum_{m\in\mathbb Z^d\setminus\{0\}} \frac{m_jm_k}{|m|^{d+2}}f(n-m), \qquad c_d=\frac{\Gamma\left(\frac{d+2}{2}\right)}{\pi^{d/2}}. \] For $j\neq k$ and every fixed $1<p<\infty$, we prove that \[ \|R_{\mathrm{dis}}^{(jk)}\|_{\ell^p\to\ell^p} = c_d\left[ \frac{2}{2^{d/2}} + \left(\frac83+o(1)\right)\frac{d}{3^{d/2}} \right] \] and \[ \frac{2c_d}{2^{d/2}} \leq \|R_{\mathrm{dis}}^{(jk)}\|_{\ell^1\to\ell^{1,\infty}} \leq c_d\left[ \frac{2}{2^{d/2}} + \left(\frac83+o(1)\right)\frac{d}{3^{d/2}} \right]. \] Since $ c_d\sim \sqrt{\pi d}\left(\frac{d}{2\pi e}\right)^{d/2} $ by Stirling's formula, thl $\ell^p$ estimates give a negative answer to the conjecture proposed by Bañuelos and Kim in \cite{BK2}. The diagonal case exhibits quite a different phenomenon: for every $j$ and every $1<p<\infty$, $R_{\mathrm{dis}}^{(jj)}$ is neither bounded from $\ell^p(\mathbb Z^d)$ to $\ell^p(\mathbb Z^d)$ nor of weak type $(1,1)$. Cancellation is restored for the operators $R_{\mathrm{dis}}^{(jj-kk)} =R_{\mathrm{dis}}^{(jj)}-R_{\mathrm{dis}}^{(kk)}$. For every fixed $1<p<\infty$, \[ \|R_{\mathrm{dis}}^{(jj-kk)}\|_{\ell^p\to\ell^p} = 4c_d\left[ 1+(1+o(1))\frac{d}{2^{d/2}} \right], \] and \[ 4c_d \leq \|R_{\mathrm{dis}}^{(jj-kk)}\|_{\ell^1\to\ell^{1,\infty}} \leq 4c_d\left[ 1+(1+o(1))\frac{d}{2^{d/2}} \right]. \]

For an integer $k \ge 0$, let $f_k(x) = 1 + e(x) + e((k+2)x)$ and $g_k(x) = 1 + e(x) - e((k+2)x)$ on $\mathbb{T} = \mathbb{R}/\mathbb{Z}$, where $e(x) = e^{2\pi i x}$. Mockenhaupt conjectured that $\|g_k\|_{L^p(\mathbb{T})} > \|f_k\|_{L^p(\mathbb{T})}$ whenever $2k < p < 2k+2$. The conjecture was previously known for $k \le 5$. We give a single analytic proof valid for every $k \ge 4$; in particular, this settles all previously open cases $k \ge 6$ and establishes the conjecture for every $k \ge 0$. The proof reduces the norm comparison to resonant Fourier coefficients on the two-torus and represents these coefficients, after analytic continuation, by triple-Bessel integrals. Neumann's product formula and the Weber-Schafheitlin formula yield a quantitative positive lower bound for the leading mode, while the remaining odd modes are controlled by a uniform tail estimate. The leading mode is then shown to dominate the tail for every $k \ge 4$. AI Usage. The mathematical argument of this paper was produced by the auto-research system Apex Math, an AI system built by Apex Intelligence. See Appendix A for the complete AI usage statement.

For the one-sided Hardy–Littlewood maximal operator $M_d^+$ on $\mathbb R^d$, the natural two-weight Muckenhoupt condition was shown by Sawyer in 1986 to characterize the weak $(p,p)$ inequality in dimension one, and by Forzani, Martín-Reyes and Ombrosi in 2011 in dimension two. At the endpoint $p=1$ in dimension three, Ombrosi and Nazarov have recently given negative answers to both the Fefferman–Stein-type question and the related weighted weak-type $(1,1)$ question \cite{OmbrosiNazarov} (personal communication). In this paper, we prove that the two-weight characterization fails for every $d\geq 3$ and $p>1$. More precisely, for every $1<p<\infty$ and every $d\geq 3$, there exist weights $w$ and $v$ such that $$ A_{p,d}^+(w,v)<\infty, \qquad \|M_d^+\|_{L^p(v)\to L^{p,\infty}(w)}=\infty. $$ The proof uses a finite two-dimensional Hardy operator whose weak operator norm is bounded below by $c_p(\log N)^{1/p'}$. This operator is embedded into the three-dimensional lattice maximal operator and then transferred to the continuous setting. A tensor extension yields the same failure in every dimension $d>3$.

For $d\geqslant3$, let $p^*=\frac{2d}{d-2}$. We prove that the Born-Jordan distribution maps $L^{2,p^*}(\mathbb R^d)\times L^{2,p^*}(\mathbb R^d)$ boundedly into $L^{p^*}(\mathbb R^{2d})$. The Lorentz index $p^*$ cannot be increased in either variable. In particular, this gives the endpoint $L^2\times L^2\to L^{p^*}$ estimate, settling the critical boundedness problem raised by Stra, Svela, and Trapasso \cite{SST26}.

Motivated by a log-convexity question of Bennett and Tao, we consider Gowers-type functionals defined by $L^p$ norms of multiple autocorrelations. We prove sharp bounds in terms of $L^q$ norms and characterise the near-extremisers on Euclidean spaces as well as locally compact abelian groups. We also establish two families of degree-lowering inequalities and study their near-extremisers. As a byproduct of this broader theory, we show that the constant in the log-convexity estimate for Gowers norms is strictly less than unity, confirming the aforementioned conjecture of Bennett and Tao. Finally, we characterise the $p$ for which the Gowers-type functionals necessarily satisfy the triangle inequality on nonnegative measurable functions.

The classical Melan equation modeling suspension bridges is considered. We first study the explicit expression and global properties of the analytical solution for the simplified “less stiff” model, based on which we derive an a priori estimate for the original classical Melan equation and establish, under an explicit load condition, that it has a unique solution with nonnegative integral under downward live loads, thereby showing the uniqueness of the corresponding deflection curve in the engineering setting. We also develop an efficient iterative approximation method by taking the solution of the simplified “less stiff” model as the first iterate, and prove its geometric convergence with explicit error estimates. The applicability and computational efficiency of the method are demonstrated through calculations for two actual bridges, which also quantify the influence of the nonlinear nonlocal term on the solution and clarify the relationship between the simplified and original models. Several engineering observations are verified and explained, and some related open problems are suggested.

We construct wavelet orthonormal bases for nonexpansive dilations and integer translations that do not admit a wavelet set. In addition, these orthonormal bases do not satisfy the Calderón sum formula. In particular, we show the existence of a wavelet basis for a dilation matrix with determinant one.

For $0<\lambda<1/2$, let $K_{\lambda}$ be the attractor of the iterated function system $\{\lambda x, \lambda x+1-\lambda\}$, and put $C_\lambda=K_{\lambda}\times K_{\lambda}$. We study the intersection \[ E_\lambda=C_\lambda\cap S^1 \] and the Minkowski sum \[ A_{\lambda}=C_\lambda+S^1, \] where $S^1$ is the unit circle. For the intersection problem, we prove that $E_\lambda$ has the cardinality of the continuum for $(\sqrt{3}-1)/2<\lambda<1/2$, and $\dim_{\rm H} E_\lambda>0$ for $\sqrt{2}-1<\lambda<1/2$. We also establish quantitative lower bounds for $\dim_{\rm H} E_\lambda$ for $\lambda$ near $1/2$; in particular, $\dim_{\rm H} E_\lambda$ approaches $1$ as $\lambda\uparrow1/2$. For the Minkowski sum problem, we prove that $A_{\lambda}$ has nonempty interior throughout the previously open range $1/4<\lambda<1/3$, answering a question of Simon and Taylor. Together with earlier results of Simon and Taylor, our theorem yields the complete classification: $A_{\lambda}$ has nonempty interior in $\mathbb{R}^2$ if and only if $1/4<\lambda<1/2$. More generally, we prove that $C_\lambda+\Gamma$ has nonempty interior for every regular $C^1$ closed curve $\Gamma$ whenever $1/4<\lambda<1/2$.

Let $R\subset\mathbb{R}^d$ be a reduced root system, $G$ the associated finite reflection group, and $k\ge0$ a $G$-invariant multiplicity function. We develop a Dunkl analogue of the obstacle/partial-balayage method of Ouyang, Spector, and Stockdale (https://arxiv.org/abs/2608.18068) for the Euclidean fractional Laplacian. For $0<s<2$, nonnegative $f\in L^1(\nu_k)\cap L^2(\nu_k)$, and $\lambda>0$, we obtain a decomposition \[ f=\mu+(-\Delta_k)^{s/2}u, \qquad 0\le\mu\le\lambda, \] with $\mu=\lambda$ on $\Omega=\{u>0\}$ and \[ \lambda\nu_k(\Omega)\le\|f\|_{L^1(\nu_k)}. \] As an application, we prove that the vector Dunkl–Riesz transform $\mathcal R_k=\nabla_k(-\Delta_k)^{-1/2}$ is of weak type $(1,1)$ with constant at most $(M_k+2)$, where \[ M_k=\#\{\alpha\in R_+:k(\alpha)>0\}. \] For $G$-invariant functions, the reflection terms vanish and the same argument gives the universal constant $2$. We further establish a dimension-free weak-type $(1,1)$ estimate for the Dunkl–Schrödinger Riesz transform.

We prove and substantially extend a conjecture of Gazeau, Josse-Michaux, and Monceau concerning the extreme positive zeros of Hermite polynomials, which arose from a finite-dimensional quantisation of the phase plane. For the generalised Laguerre polynomial $L_m^{(\alpha)}$ of degree $m$ and parameter $\alpha>-1$, the product of its $j$th smallest and $j$th largest zeros is strictly increasing with $m$ for each fixed positive integer $j$, once $m\ge2j-1$. The classical Hermite-Laguerre identities then prove the original conjecture for the degree-$N$ Hermite polynomial $H_N$, in both parities and for every $N\ge4$. As a further consequence, a one-parameter extension involving a deformation of the corresponding Jacobi matrix is settled.

For every $1<p\le\infty$, we prove dimension-free maximal inequalities over all radii for normalized averages over Euclidean balls in $\mathbb Z^d$. In particular, this settles the $\ell^2$ question attributed to Stein. The proof uses a two-saddle expansion at integer squared radii to compare ball multipliers with normalized discrete Gaussians at the zero and parity frequencies. First-order estimates give the full-radius $\ell^2$ bound. For $1<p<2$, we combine higher-order residual estimates with dimension-uniform $\ell^1$ bounds for the residuals and levelwise interpolation to obtain the full range.

For the power mean $P_t$, $t<1$, let \[ \Lambda_N(t)=\sup_{x_k>0} \frac{\sum_{n=1}^N P_t(x_1,\ldots,x_n)}{\sum_{n=1}^N x_n} \] be its finite Hardy constant. Negative powers, the geometric mean, and the usual positive-exponent Hardy inequality correspond respectively to $t<0$, $t=0$, and $0<t<1$. With the analytic parameter $q=t/(1-t)\in(-1,\infty)$, and writing $\lambda_N(q)=\Lambda_N(q/(1+q))$, we derive one exact scalar shooting map for all three regimes; its apparent singularity at $q=0$ is removable and gives the finite Carleman problem exactly. The map has a common terminal condition and its limiting vector field has a quadratic critical bottleneck at \[ y_*(q)=1+q,\qquad \Lambda_*(q)=(1+q)^{(1+q)/q}. \] We prove that there is a unique real-analytic phase $\kappa(q)$ for which the finite-defect expansion has zero cubic coefficient and, uniformly for $q$ in compact subsets of $(-1,\infty)$, for every fixed $L\ge2$, \[ \Lambda_*(q)-\lambda_N(q) =\sum_{j=2}^{L}\frac{A_j(q)}{(\log N+\kappa(q))^j} +O\!\left((\log N)^{-L-1}\right). \] In this phase-normalized scale, \[ A_2=2\pi^2(1+q)\Lambda_*,\qquad A_3=0,\qquad A_4=-\frac{\pi^2}{3}(2q^2+5q+5)A_2. \] Thus the classical Carleman and positive-Hardy leading corrections are projections of a single analytic family. At the Carleman parameter we obtain $A_2(0)=2e\pi^2$ and $A_4(0)=-(10/3)e\pi^4$; the latter is also recovered by a direct local residue calculation. The phase itself contains the global discrete defect. After phase normalization, the first coefficient involving a nonconstant discrete response is $A_5$, at order $(\log N+\kappa(q))^{-5}$.