Papers

We construct a two-parameter family of polynomial vector fields in $\mathbb{R}^3$ and prove that, on a nonempty open set of parameter values, every member of the family is Hurwitz and possesses two non-constant periodic orbits, one of saddle type and one attracting. This answers Problem 19 in Gasull's list of open problems on low-dimensional dynamical systems concerning the existence of non-constant periodic orbits in Hurwitz smooth vector fields on $\mathbb{R}^3$. This also provides counterexamples to the expected non-existence of non-constant periodic orbits for Hurwitz polynomial vector fields; a question that arose from the first counterexamples to the Markus-Yamabe conjecture in the late 1990s and that was stated by Meisters in his featured MathSciNet review of the Bernat-Llibre counterexample.

We prove a spectral multiplier theorem for the standard sub-Laplacian on the Engel group under a scale-invariant local $H^s$ condition with $s>2$. The multiplier operators $F(\mathcal L)$ are of weak type $(1,1)$ and bounded on $L^p$ for every $1<p<\infty$. Together with the known lower bound, this identifies the sharp threshold as $2$, half the topological dimension. The proof combines a weighted Plancherel estimate for quartic oscillators with a decomposition into rank-two spectral subspaces, which controls parameter derivatives uniformly as one well escapes to infinity. For fixed smooth multipliers compactly supported in $(0,\infty)$, we also determine the leading term and first correction for the localized kernels. Whether the weak type $(1,1)$ estimate holds under the local $H^2$ condition remains open.

We prove a sharp $L^\infty$ bound for the density of an infinite convolution in terms of $L^2$ norms of convolution powers of the summand densities. The bound may be finite even when every summand density is unbounded and the products arising from Young's inequality diverge.

Using differential Galois theory—specifically, Magid's theory of the complete Picard-Vessiot closure—we show that a system of first-order differential equations in $n$ variables admits a Liouvillian first integral if and only if it admits a first integral in a Liouvillian Picard-Vessiot extension, if and only if it admits a first integral in its Picard-Vessiot ring. Consequently, a system with a Liouvillian first integral admits one in a differential field obtained from the field of rational functions by first taking a finite algebraic extension, then adjoining an exponential of an integral, and then an integral. This yields a new proof of Singer's theorem on Liouvillian first integrals of autonomous planar systems (Trans.\ Amer.\ Math.\ Soc.\ 333, 1992) and of its recent extension to autonomous systems in $n$ variables by Aziz et al. (arXiv:2512.15522). Our results hold over any differential field of characteristic zero with an algebraically closed field of constants; in particular, they apply to non-autonomous systems. Finally, we exhibit a system that admits a first integral in a non-Liouvillian Picard-Vessiot extension but none in its Picard-Vessiot ring and thus establishing that the Liouvillian hypothesis cannot be dropped.

In this paper, we investigate the Takagi-van der Waerden function, $$ T_r(x) = \sum_{n=0}^{\infty} \frac{\phi(r^n x)}{r^n} ,\quad x\in [0,1], \quad r \geq 2, $$ where $\phi(x)={\rm dist}(x,\mathbb{Z})$ represents the distance from $x$ to the nearest integer. Lagarias and Maddock[13] introduced the notion of local level sets for the classical Takagi function $T_2$. They proved that if $y$ is uniformly distributed over the range of $T_2$, then the expected number of local level sets contained in the level set at height $y$ equals $3/2$. We define the corresponding local level sets for every integer $r\geq2$ and prove that, if $y$ is uniformly distributed over the range of $T_r$, then the expected number of local level sets contained in the level set at height $y$ equals $(r+1)/r$ if $r$ is even, and $(r+1)/(r-1)$ if $r$ is odd.

We characterize when a banded totally positive matrix admits a positive bidiagonal factorization. In finite dimension such a factorization always exists. For semi-infinite matrices of arbitrary finite bandwidth, existence is equivalent to convergence of a finite family of positive series determined by the Metelmann factors. Their reciprocals are the limiting values of scalar Jacobi continued fractions. We also prove that every bounded banded totally positive matrix can be approximated in operator norm by matrices admitting positive bidiagonal factorizations, through perturbations confined to a fixed initial block. For banded Toeplitz matrices, the negative roots of the polynomial symbol give a global factorization with bounded factors. Ordered partitions of these roots determine explicit mixed-type multiple orthogonal polynomials. Applying the existing spectral Favard theorem, we compute their matrix of positive measures: its support is one interval, and the measure is purely absolutely continuous with an algebraic density of rank one. These formulas also give transition and first-return laws for the associated killed random walks.

Let $(M,g)$ be a compact $d$-dimensional Riemannian manifold without boundary, and let $\{e_j\}_{j=0}^\infty$ be an orthonormal basis of Laplace eigenfunctions with frequencies $\{\lambda_j\}_{j=0}^\infty$. For a domain $\Omega\subset M$, consider $$ N_\Omega(\lambda) = \sum_{\lambda_j\leq\lambda} \left|\int_\Omega e_j\,dV_g\right|^2. $$ Suppose that $\partial\Omega$ is $s$-Ahlfors regular, with $s\in{[d-1,d)}$, and that $\Omega$ satisfies a two-sided corkscrew condition. We prove that $$ N_\Omega(\lambda) = \operatorname{vol}(\Omega)+O\bigl(\lambda^{s-d}\bigr), $$ and that this remainder is sharp. We further identify the geometric quantity governing the second term: an exact asymptotic of order $\lambda^{s-d}$ holds precisely when the boundary admits a natural $s$-dimensional crossing volume $\mathcal V^s(\partial\Omega)$, in which case $$ N_\Omega(\lambda) = \operatorname{vol}(\Omega) - C_{d,s}\, \mathcal V^s(\partial\Omega)\, \lambda^{s-d} + o\bigl(\lambda^{s-d}\bigr). $$ For admissible self-similar boundaries, the crossing volume exists in the non-lattice case, and hence the above two-term asymptotic holds. In the lattice case, the second-order behavior is instead governed by a log-periodic profile.

We develop a sparse domination framework adapted to nested families in general measure spaces, possibly including point masses. We show that the Carleson condition is equivalent to a slight generalization of the usual sparse condition, and use this characterization to obtain one- and two-weight estimates for the associated averaging operators, together with converse testing conditions. We apply our general results to integral operators on locally finite trees. Trees come naturally equipped with two nested geometries: sectors and pruned sectors. We prove pointwise sparse domination of natural versions of the harmonic Bergman projector by the corresponding averaging operators, depending on whether the measure on the tree is doubling or not, and show that the distinction is necessary in general. For the positive Bergman projector, we obtain pointwise comparability. As a consequence, we obtain new weighted inequalities and boundedness results for the Bergman projector and a class of Toeplitz-like multipliers.

We study weighted coefficient estimates for the Laguerre transform in terms of moduli of smoothness generated by the Laguerre translation. For every integer $r\ge1$ and $1\le p\le2$, a weighted $\ell^{p'}$ norm of $\min\{1,nt\}^{r}\widehat f_\alpha(n)$ is bounded by the modulus of order $r$; for $2\le p\le\infty$ the reverse direction holds on the same weighted sequence scale. At $p=2$ these estimates yield a two-sided equivalence, and testing individual Laguerre polynomials proves optimality of the multiplier weight up to constants. The proof combines Hausdorff–Young inequalities with an averaged lower bound for the translation multiplier. As consequences we recover Jackson- and Bernstein-type estimates and the known $L^2$ approximation characterisation of the Lipschitz classes for $0<\beta<r$. We also give the corresponding one-sided coefficient-tail criteria for general $p$, together with saturation and endpoint statements.

We prove that the centered Hardy–Littlewood maximal operator is not of weak type $(p,p)$ for any $1\le p<\infty$ in three solvable settings: the lamplighter group $(\mathbb{Z}/2\mathbb{Z})\wr\mathbb{Z}$ with the switch–walk–switch word metric, the Baumslag–Solitar groups $BS(1,n)$, $n\ge2$, with their digit word metrics, and the three-dimensional Sol group with any left-invariant Riemannian metric and its Riemannian volume. To the best of our knowledge, we give the first examples of finitely generated groups, equipped with word metrics and counting measure, for which the centered Hardy–Littlewood maximal operator fails to be of weak type $(1,1)$.

We prove that, for all $x > 0$, the product of modified Bessel functions satisfies \[ \frac{1}{2\sqrt{x^2+\nu^2+1/5}}<I_\nu(x)K_\nu(x)<\frac{1}{2\sqrt{x^2+\nu^2-1/4}} , \] where the lower bound holds for $\nu>-1$ and the upper bound for $\nu\ge 1/2$, and that the sum of the logarithmic derivatives satisfies \[ -\frac{x}{x^2+\nu^2-1}<\frac{I_\nu'(x)}{I_\nu(x)}+\frac{K_\nu'(x)}{K_\nu(x)}<-\frac{x}{x^2+\nu^2+7/20}, \] where the upper bound holds for $\nu>-1$ and the lower bound for $\nu\ge 1$. All four constants are best possible. The four proofs share the same elementary structure: in each case the relevant auxiliary function satisfies a second-order differential inequality, and the inequality follows from the strong maximum principle. Among other applications, we obtain sharp monotonicity properties of the product and improved bounds for the ratios $I_{\nu-1}(x)/I_\nu(x)$ and $K_{\nu+1}(x)/K_\nu(x)$.

We prove Mizohata–Takeuchi-type estimates for the Fourier extension operator associated with the truncated hyperbolic paraboloid in $\R^3$. Our main result controls the weighted $L^2$ norm of the extension operator by the $L^2$ mass of the weight over a multiscale family of parallelepipeds arising from $\ell^2$ decoupling for the hyperbolic paraboloid. We deduce this result from a weighted $L^2$ estimate for functions whose Fourier transforms are supported in a small neighborhood of the hyperbolic paraboloid, which we establish using a bilinear refined decoupling estimate.

We study sampling discretization of the uniform norm for trigonometric polynomials with frequencies in a hyperbolic cross of level $N$ on $\T^d$. For every $d,N\ge2$ and $\varepsilon\in(0,1)$, we construct an explicit norming set with at most $\bigl(C_1(1+\varepsilon^{-1})\bigr)^{d-1}N^{1+\varepsilon}$ points and norming constant at most $\bigl(C_2(1+\varepsilon^{-1})\bigr)^{d-1}$, where $C_1,C_2$ are absolute constants. We also prove that independent Haar-distributed points achieve the same exponent $1+\varepsilon$: a sample of size at least $C(d,\varepsilon)N^{1+\varepsilon}\log(2/\eta)$ norms the entire space simultaneously with probability at least $1-\eta$, with a norming constant bounded by $\exp(C_d\varepsilon^{-d})$ and independent of $N$. The deterministic construction combines uniformly stable de la Vallée Poussin sampling operators with a Smolyak-type combination identity. The random result follows from an abstract norming theorem for sums of spaces associated with commuting regular partitions, together with a multiscale approximation of hyperbolic-cross polynomials. Finally, we show that every norming set of constant $B>1$ for the trigonometric polynomials with frequencies in $\{-1,0,1\}^d$ has at least $\bigl(\pi d/(2e\log B)\bigr)^{d/2}$ points. This rules out quadratic bounds in the dimension of the polynomial space with a prefactor growing only polynomially in $d$ when the norming constant is fixed.

Let $m,n\geq2$ be integers. We study the least exponent $t_{m,n}$ such that the $m$-fold additive energy of any subset $A$ of the discrete cube $\{0,1,\cdots,n-1\}^d$ in any dimension $d$ satisfies $E_m(A)\leq |A|^{t_{m,n}}$. For every fixed $m$, we obtain the asymptotic formula $$ t_{m,n}=2m-1-(1+o(1))\log_n \frac{(2m-1)^{m-1/2}}{(2m)^{m-1}} \qquad(n\to\infty). $$ For $m=2$, this gives $$ t_{2,n} = 3 - (1+o(1)) \log_n\frac{3\sqrt{3}}{4}, $$ proving a conjecture of the first author.

We prove McLeod's (1959) conjecture that $(h_{\ell+r}/h_\ell)^{1/r}$ is subadditive for all $\ell\ge0$ and $r\ge1$, where $h_k$ is the complete homogeneous symmetric polynomial of degree $k$ (J. B. McLeod, "On four inequalities in symmetric functions", Proc. Edinburgh Math. Soc., 1959). We first prove that the ratio $h_k/h_{k-1}$ is convex, and our proof takes an unexpected detour through queuing theory. The geometric mean of $r$ consecutive such ratios equals McLeod's conjectured function, but geometric means of convex functions need not be convex. Our second and decisive step is a tensorization trick that couples neighboring ratios tightly enough to make their geometric mean convex.

Consider a system of equations $A(x)\cdot F(x) = B(x)$ on a subset $X$ of $\mathbb R^n$, where $A(x)$ and $B(x)$ are matrix- and vector-valued semialgebraic functions on $X$, and the unknown $F(x)$ is a field of vector-valued $m$-jets. We assume there is a solution which is the field of Taylor polynomials of order $m$ on $X$ of a $\mathcal C^m$ vector-valued function $f$ on $\mathbb R^n$, and ask whether we can find a $\mathcal C^m$ semialgebraic solution $f$. Our main result is a positive answer in the case $\dim X = 1$. The methods are based on an article of Fefferman and Luli for $X \subset \mathbb R^2$, and a secondary goal is to show that their approach applies in a much simpler way in the case $\dim X =1$, or in the case of the semialgebraic Whitney extension problem in $\mathbb R^2$. Our results hold more generally for functions definable in an o-minimal expansion of $\mathbb R$.

A synthetic view is provided on scattered results which were never presented in this form in the literature. First we investigate precisely the oscillation length for periodic functions. Then, we establish the existence of a uniform oscillation lentgh for any scalar almost periodic function with mean-value 0. After that, we investigate upper bounds of the oscillation length for some almost periodic functions which appear in the study of continuum mechanics, and we show the existence of a finite oscillation length with respect to time in any open subset of some vibrating continuous media. The paper finally recalls a long standing open problem on the vibrations of a square membrane with fixed edge.

We prove a noncommutative maximal ergodic inequality for averages along polynomial sequences. Let $\gamma$ be a trace-preserving automorphism of a semifinite von Neumann algebra $(\mathcal N,\tau)$. We show that the associated polynomial averages \begin{equation*} A_Nf:=\frac1N\sum_{n=1}^N\gamma^{P(n)}(f), \qquad N\in\mathbb N, \end{equation*} satisfy a strong maximal inequality on $L_p(\mathcal N)$ for every $1<p<\infty$, extending the previously known restricted range of $p$. The proof follows Bourgain's major-arc strategy but requires substantially new ideas and tools for operators that may have further applications in noncommutative analysis. More precisely, we obtain a localized maximal inequality by developing a noncommutative version of Stein's extrapolation, novel combinatorial methods, and a surprising multilinear version of Doob's maximal inequality. For the required decaying $L_2$-approximation, we combine two of Bourgain's constructions in a way that avoids the multi-frequency maximal inequality used in the scalar proof.

We obtain improved Fourier restriction and Kakeya maximal estimates in $\mathbb R^4$. More precisely, by proving the two-ends incidence estimates $\mathbf{TE}\left(\frac{133}{44},\frac{177}{88},\frac34\right)$ and $\mathbf{TE}\left(\frac{386453}{125988},\frac{386453}{125988},\frac{358951}{377964}\right)$, we extend the restriction range to $p>2+ \frac{176}{221}$ and the Kakeya maximal estimate up to dimension $3.06737$. The implications to the known lower-bounds for the Hausdorff dimension of Kakeya sets in $\mathbb{R}^4$ and the Bochner–Riesz conjecture are also studied.

The goal of the present paper is to investigate the exact decay rate of the tail probability $\mathbb{P}(|X_1-x_0|>R)$ for large $R$, where $X_t$ is the solution to a multidimensional stochastic differential equation driven by a fractional Brownian motion with initial condition $x_0$. In the first place, under the assumption of uniform ellipticity, we establish a general $(2H+1)$-Weibull lower tail estimate in Young's regime of $H\in(1/2,1)$ and a general Gaussian lower tail estimate in the rough regime of $H\in(1/4,1/2)$. These two estimates are seen to be sharp in their respective regimes. In the second place, we prove a striking fact by constructing explicit examples that a uniformly elliptic system with $C_b^\infty$-coefficients could fail to have Gaussian lower tail in Young's regime. As a consequence, in our modest opinion, the multidimensional Gaussian lower estimate of \cite{BKT16} might not hold in its current form of generality without further assumptions on the vector fields. In the third place, we provide a simple nondegeneracy condition on the vector fields, under which a Gaussian lower tail can be established in Young's regime. In addition, given the existence of the aforementioned counterexamples, we prove another striking fact that any $2\times 2$ periodic, uniformly elliptic system always has Gaussian lower tail in Young's regime. Lastly, in the rough regime we establish a general $(2H+1)$-Weibull lower tail for a rich class of systems that satisfy a noncommutativity condition on the vector fields. This result provides a partially affirmative answer to a conjecture raised in \cite{BG24} on the genericness of the well-known Cass-Litterer-Lyons estimate for noncommutative systems.

In this paper, we give the complete characterization of the boundedness of the discrete fractional maximal operator $M_\gamma $, defined on $\mathbb{Z}^n$, $n\in\mathbb{N}$, $\gamma\in [0,n)$, between the classical Lorentz sequence spaces $\lambda_p(v)$ and $\lambda_q(w)$, where $0<p,q<\infty$, $\{v(m)\}$ and $\{w(m)\}$ are non-negative sequences defined on $\mathbb{N}$. We first obtain a sharp upper estimate for the nonincreasing rearrangement of the discrete fractional maximal operator $M_\gamma x$, with a supremal operator involving a discrete Hardy operator and the nonincreasing rearrangement of the sequence $\{x(k)\}$, and we show that this estimate is sharp by showing a lower estimate for special sequences. Using this result, we can reduce the characterization of the boundedness of the discrete fractional maximal operator $ M_\gamma$ between the discrete weighted Lorentz sequence spaces to the boundedness of the discrete supremum operator between weighted Lebesgue sequence spaces, restricted to nonincreasing sequences. This problem was investigated in the authors' recent paper, which enables us to obtain the final characterization.

Let $w \colon (0,1) \to \mathbb{R}_{+}$ be a weight. We prove that for an arbitrary Schauder basis $\{e^{i \lambda_n t}\}_{n \in \mathbb{Z}}$ in $L^2(0,1)$ and an arbitrary lacunary defect set $A \subset \mathbb{Z}$ the system $\{w(t) r_n(t)\}_{n \in \mathbb{Z} \setminus A}$ is always complete and never minimal for any weight $w$ satisfying a natural decay condition, which is sharp on the exponential scale. Moreover, we establish a simple combinatorial criterion for systems of the form $ \{ w(t) e^{2\pi i n t} \}_{n \in \mathbb{Z} \setminus A} $ to be complete and minimal in $L^2(0,1)$ for an arbitrary weight $w$ and a finite defect set $A \subset \mathbb{Z}$.

Boscaggin, Feltrin and Zanolin conjectured that the Neumann and periodic problems associated with the equation \begin{equation*} u”+a(t)u^\gamma=0 \end{equation*} with a two-step indefinite weight $a(t)$ have at most one positive solution for every $\gamma\in\mathbb{R}\setminus\left\{-1,0,1\right\}$. We prove this conjecture and complete the uniqueness picture by also including the logarithmic potential case $\gamma=-1$. More generally, for every $p>1$, we consider the equation \begin{equation*} \left(\left\lvert u'\right\rvert^{p-2}u'\right)'+a(t)u^\gamma=0 \end{equation*} under Neumann or periodic boundary conditions and establish a complete existence and uniqueness classification for all $\gamma\in\mathbb{R}\setminus\left\{0,p-1\right\}$. The necessary mean-value condition $\gamma\int_{0}^{T}a(t)\,\mathrm{d} t<0$ is also sufficient for existence when $\gamma\leq-1$ or $\gamma>p-1$, while an additional sharp condition on the mean of the weight is required in the intermediate range $\gamma\in(-1,p-1)\setminus\{0\}$. Finally, a numerical example provides evidence of multiple positive solutions for the Neumann problem whose weight has the same single change of sign but a non-constant internal profile, highlighting the role of the two-step assumption.

We establish transference results for multilinear Fourier multiplier operators from Euclidean spaces to product spaces of the form $\mathbb R^{d_1}\times\mathbb T^{d_2}$. We prove strong and weak-type transference principles that preserve the corresponding Euclidean operator bounds. As applications, we obtain bounds for certain multilinear operators, as well as for some multiplier operators arising naturally in PDEs.

We study continuous Brascamp–Lieb inequalities associated with isotropic measures on the sphere $S^{n-1}$ and on the Grassmannian $G(n,k)$. We give two proofs of the continuous Grassmannian inequality. The first proof approximates the continuous isotropic measure by discrete isotropic measures and then applies the equal-rank geometric Brascamp–Lieb inequality. In the rank-one case, the construction of the discrete isotropic measures follows from Barthe's work, which is standard, whereas in the higher-rank case the construction of finite approximate isotropic measures is given by a Tchakaloff-type theorem. Weak convergence, geometric Brascamp–Lieb inequality and Fatou's lemma then yield the continuous inequalities. The second proof generalizes the heat-semigroup preservation method of Barthe and Huet to isotropic measures on $G(n,k)$. We prove a heat-semigroup preservation theorem in the Grassmannian setting and show that this theorem yields a Grassmannian Brascamp–Lieb inequality. At the end, we also give a sketch for the mixed-rank case of the continuous Brascamp–Lieb inequality.

Let $N\ge1$, $p\in[1,\infty)$, and $\Omega\subset\mathbb R^N$ be a bounded open interval when $N=1$ or a bounded Lipschitz domain when $N\ge2$. For any given $\gamma\in\mathbb R\setminus\{0\}$ and $\lambda\in(0,\infty)$, consider the weak-type nonlocal functional \begin{align*} F_{\lambda,p,\gamma}(u;\Omega):=\lambda^p \underset{|u(x)-u(y)|>\lambda|x-y|^{1+\frac{\gamma}{p}}} {\int_\Omega\int_\Omega}|x-y|^{\gamma-N}\,dy\,dx. \end{align*} In this article, we prove that, for $\gamma\in(0,\infty)$ as $\lambda\to\infty$ or for $\gamma\in(-\infty,-1]$ as $\lambda\to0^+$, the family $F_{\lambda,p,\gamma}(\cdot;\Omega)$ converges, in the sense of $\Gamma$-convergence in $L^p(\Omega)$, to the functional \begin{align*} \Psi_{p,\gamma}^{\Omega}(u) :=\begin{cases} C_{N,p,\gamma}^cell\displaystyle\int_\Omega|\nabla u|^p\,dx, &p\in(1,\infty)\ \hbox{and}\ u\in W^{1,p}(\Omega),\\[2mm] C_{N,1,\gamma}^cell|Du|(\Omega), &p=1\ \hbox{and}\ u\in BV(\Omega),\\[1mm] \infty,&\hbox{otherwise}. \end{cases} \end{align*} where the positive constants $C_{N,p,\gamma}^{\mathrm{cell}}$ are independent of $\Omega$. As applications, we further show that, for $\gamma\in(0,\infty)$ as $\lambda\to\infty$ or for $\gamma\in(-\infty,-1]$ as $\lambda\to0^+$, the weak-type nonlocal functional $F_{\lambda,p,\gamma}(\cdot;\mathbb R^N)$ converges, in the sense of $\Gamma$-convergence in $L^1_{\rm loc}(\mathbb R^N)$, to a similar functional; this gives an complete answer to the problem posed by Brezis et al. in [Section~7C, Anal. PDE 2024].

Let $E\subset\mathbb R^2$ be analytic with $t=\dim_{\mathrm H}E>1$. We prove $\dim_{\mathrm H}\Delta E\ge 4/5+\eta_0$ for an absolute, non-explicit constant $\eta_0>0$, and \[ \dim_{\mathrm H}\Delta E\ge \frac{t^2-2t+5}{2t^2-7t+10} \quad\text{for }1<t\le\frac{11-\sqrt{41}}{4}, \] which reaches $7/8$ at the right endpoint. For pinned distances, $\dim_{\mathrm H}\Delta_xE\ge q_*=0.7813\ldots$ for every $x\in\mathbb R^2$ outside an exceptional set of Hausdorff dimension at most one, with stronger bounds depending on $t$ and further exceptional-set estimates. If $E$ is compact and supports an $s$-Frostman measure $\mu$, $s>1$, then \[ \underline{\dim}_{\mathrm B}\Delta_xE\ge \max\left\{\frac45+\eta_0,\min\left\{\frac{3s+1}{5},\frac78\right\}\right\} \] for $\mu$-almost every $x$. The same bound holds for pinned Hausdorff dimension under a common dyadic mass law, allowing oscillation with scale. We retain information about both endpoints in the Kolmogorov complexity estimates of Stull and Fiedler–Stull. Applying the partition estimates with the endpoints reversed yields $4/5$ and the rational bound. Equality forces both information profiles to have excess information on a common interval of scales. Encoding one source measure for all pins converts this constraint into a configuration of points and circles with many rich perpendicular bisectors. Lower bounds of Katz–Tardos and Ren–Wang contradict the upper bound of Fu–Gan–Ren; compactness then gives $\eta_0>0$. This rigidity argument has a precedent in the work of Héra, Shmerkin and Yavicoli on Furstenberg sets, using Bourgain's projection theorem. A separate finite-scale selection proves the pinned lower box bound.

Recently, it was shown in arXiv:2607.27629 that under a very general condition, a generic family of curved Kakeya sets in $\mathbb{R}^3$ not trivially contained in a hypersurface has Hausdorff dimension strictly greater than 2. In this paper we leverage large language models to give an explicit construction of a family of curves failing the worst compression phenomenon in arXiv:2607.27629, but whose union has Hausdorff dimension and upper box dimension $5/2$. The example shows that there are Kakeya compression phenomena besides the trivial example of concentrating near a hypersurface. This answers a question raised in section 1.9 of arXiv:2607.27629. The models used are ChatGPT 5.6 Sol and ChatGPT 6 Astra.

We determine the sharp uniform constant in the quartic stability estimate for the logarithmic Sobolev inequality on finite cycles. We equip $C_{n}=\mathbb{Z}/n\mathbb{Z}$ with the uniform probability measure, and we let $\mathcal{E}_{n}$ be the Dirichlet form of the simple random walk on $C_{n}$, normalized so that its spectral gap is one. Frank and Ivanisvili recently proved the sharp logarithmic Sobolev inequality $2\mathcal{E}_{n}(u)\geq \mathrm{Ent}(u^{2})$ for all $n\geq 4$. We prove that $2\mathcal{E}_{n}(u)-\mathrm{Ent}(u^{2})\geq \frac{1}{3}\Vert u-1\Vert _{2}^{4}$ for all $n\geq 4$ and all nonnegative $u$ with $\Vert u\Vert _{2}=1$. This is the sharp form, uniformly in $n$, of the quartic stability estimate of Xie and Zhang, who obtained the constant $\frac{1}{12}$. The same estimate holds on the circle and for the word-length energy on $C_{n}$. Among these models, the four-cycle is the only extremal one: on $C_{4}$ the constant $\frac{1}{3}$ is sharp but not attained, while on $C_{n}$ with $n\geq 5$ and on the circle the sharp constant is at least $\frac{1}{3}+c$, where $c>0$ does not depend on $n$. We also give explicit upper bounds for these sharp constants. Our second main result is a stability estimate for the cubic Sobolev inequality of Frank and Ivanisvili, which is the main step in their proof. It gives the quantitative version of this inequality expected by Frank and Ivanisvili, with the optimal powers of $\Vert u-1\Vert _{2}$: four on $C_{n}$ with $n\geq 5$ and on the circle, and six on $C_{4}$. Two key new ingredients of the proofs are a bound for the first Fourier coefficient of $u$ in terms of its mean, which holds because $u$ is nonnegative, and a sharp stability estimate for Gross's two-point logarithmic Sobolev inequality, with sharp constants in both the fourth and the sixth powers of the distance.

Let $\mu$ be a doubling Radon measure in $\mathbb R^d$ with support $X$. We prove a local and quantitative form of the Bennewitz–Lewis theorem in this setting: if a Radon measure $\nu$ satisfies the Bennewitz–Lewis condition on a bounded relatively open set $V\subset X$ with a small enough constant, then $\nu|_V\ll\mu|_V$ and its density satisfies a reverse Hölder inequality on every compact subset $K\subset V$. Along the way we develop the dyadic toolkit these arguments run on — triple cubes defined through neighbours, the basis $\mathcal{D}\cup3\mathcal{D}$, and a Whitney decomposition — and we collect the Muckenhoupt-type classes of weights over that basis. The same machinery yields a local Gehring lemma for enlarged balls: both theorems are deduced from a single bootstrapping lemma, stated once for $p=1$ and $p>1$.