Papers

We establish product-profile anti-concentration, density, and Remez estimates for multi-affine polynomials on the cube. Let $X$ be uniformly distributed on $[0,1]^n$, and let $P:[0,1]^n\to\R$ be a nonconstant multi-affine polynomial of exact degree $d$, with all variables active. For a partition $\cB$ of the variables such that no monomial contains two variables from the same block, define $$ w_d(\cB) = \sum_{\cI\subseteq\cB, \abs{\cI} = d-1} \left(\prod_{B\in\cI}\sqrt{\abs B}\right) \sqrt{\sum_{C\in\cB\setminus\cI}\abs C}. $$ We prove the all-center small-ball estimate $$ \sup_{u\in\R} \Prob\{\abs{P(X)-u}\le\rho\} \le \Phi_d\left( C_dw_d(\cB)\frac{\rho}{\osc(P)} \right), $$ where $\osc(P)$ is the length of the range of $P$ and $\Phi_d(t) = t\sum_{j = 0}^{d-1}\log^j(1/t)/j!$ on $(0,1]$, capped at one. Every multi-affine polynomial admits the singleton partition, which yields the universal scale $n^{d-1/2}$; an admissible partition into at most $q$ blocks yields the improved scale $q^{(d-1)/2}n^{d/2}$. If $\cB$ has exactly $d$ blocks, then $$ w_d(\cB) = d\prod_{B\in\cB}\sqrt{\abs B}, $$ and products of centered block averages show that both the small-ball profile and the dependence on the full block-size vector are optimal, up to constants depending only on $d$. We also prove that $P(X)$ has a density $f_P$ satisfying $$ \norm{f_P}_{L^p(\R)} \le C_dp^{d-1} \left( \frac{w_d(\cB)}{\osc(P)} \right)^{1-1/p}, \quad 1<p<\infty, $$ with matching $p$- and block-scale growth for $p\ge2$ on the block-product models. As an application, we derive translation-invariant quotient Remez inequalities with the same structural scale. The proof combines weighted block selection, affine cube slicing, degree-lowering contractions, and the exact recursion underlying the product profile.

This paper studies axially symmetric Abrikosov-Nielsen-Olesen vortices in the quartic Abelian Higgs model as the magnetic flux becomes arbitrarily large. The emphasis is on developing the rigorous asymptotic analysis behind the formal large-flux expansions of Hakim, Lemaitre, and Mallick '01 and Dumitrescu, Gaikwad '25. For every coupling constant less than 4 we prove that the radius of the magnetic core is of the size predicted by the physics literature. We identify the interface between then normal and super-conducting regimes with the classical normal-to-superconducting transition layer in Ginzburg-Landau theory. The theorem of Chapman-Howison-McLeod-Ockendon '91 proves its existence, uniqueness and monotonicity. The connection of the Gaussian tail of the scalar field on the left with the interface is made directly with the Weber equation. The core Higgs equation is treated as a semiclassical Sturm-Liouville problem in the left overlap region, where the Higgs field decays like a Gaussian. The exterior region is described perturbatively around modified Bessel functions. A key step is to show that the matching map between the admissible Cauchy data at the two ends of the linearized interface equations has maximal rank.

We study the Hausdorff dimension of projections of level sets of Lyapunov exponents for affine iterated function systems. Assuming that the tuple of linear parts is typical, we obtain a variational formula in terms of pressure, topological entropy, and the Lyapunov dimensions of invariant measures corresponding to Lyapunov exponents. In $\mathbb{R}^3$, we show that this value equals the Hausdorff dimension of the canonical projection of the level set under the strong open set condition. We further extend the result to arbitrary dimensions, showing that the same formula holds for Lebesgue-almost every choice of translation vectors under a suitable contraction assumption.

We prove that there is an absolute constant $c>0$ such that for every finite $(\alpha,M)$-Ahlfors–David regular set $A\subseteq[N]$ with $0\leq\alpha<1$ and $M\geq2$, we have the sumset estimate \[|A+A|\geq |A|^{1+c(1-\alpha)/\log M}.\] We also prove a corresponding statement for Ahlfors–David regular sets in $[0,1]$. The proof combines a multiscale entropy decomposition with inverse results from additive combinatorics, showing that Ahlfors–David regularity forces a definite entropy gain at each scale. The dependence $1/\log M$ in the exponent is of optimal order.

We introduce an intrinsic notion of p-roughness for continuous paths,for p>1, defined by the uniform convergence of discrete p-energies over all shifted sufficiently fine uniform grids. We prove that this self-averaging property is equivalent to mesoscopic cancellation of the coarse-graining error for discrete p-energy, yielding a characterization that can be verified on a single uniform multiresolution. $p$-roughness refines the finite p-th variation property and implies the invariance of p-th variation across a class of partition sequences. We prove that Brownian motion is almost surely 2-rough and that fractional Brownian motion with Hurst parameter $H$ is almost surely $1/H$-rough. We also derive criteria for p-roughness based on Faber-Schauder coefficients. These results yield partition-robust formulations of higher-order pathwise calculus and energy occupation measures. Finally, we interpret coarse-graining as a renormalization flow for the p-energy and show that the p-roughness class is stable under critical time-amplitude scaling, with linear p-energy profiles as fixed points.

We prove $L^p$ bounds for multilinear multipliers whose symbols are singular along degenerate subspaces. When the singularity has a sufficiently mild degeneracy, we show that the wavepacket analysis of the nondegenerate theory can be adapted to this setting. The admissible range of exponents depends on the geometry of the singularity.

We study the expected number of zeros of the Short-Time Fourier Transform (STFT) with a Gaussian window for signals degraded by complex Gaussian colored noise. We provide an exact formula for this quantity and investigate its asymptotic behavior. From a computational perspective, we propose an algorithmic method to recover a smoothed power spectral density (PSD) directly from the spatial distribution of the transform's zeros. Our results formalize the commonly-held heuristic that detection algorithms based on spectrogram zeros, though designed for white noise, also perform adequately under moderately colored noise.

In this article, we prove the sharp Strichartz estimates on the one-dimensional torus.

We complete the characterization of the $L^p$ boundedness of the commutators $[b, H_\gamma]$ of pointwise multiplication and the Hilbert transform along monomial curves in any dimension. This fully solve a key point left open by Bongers, Guo, Li and Wick, which was affirmatively answered recently only for dimension two. We achieve this by a novel but elementary argument, avoiding the complicated geometry structure in higher dimensions.

We give negative answers to Kolmogorov's rearrangement problem and Garsia's conjecture. We construct a complete uniformly bounded orthonormal system for which every rearrangement admits a square-summable series divergent almost everywhere. The finite construction uses two copies of the trigonometric system in different orderings. The main ingredient is a combinatorial lemma which guarantees a prescribed ordering along an arithmetic progression in at least one of two related permutations. Its proof uses Szemerédi's theorem and a counting argument. A Walsh variant gives $N$-term $\{\pm1\}$-valued systems with $L^2$ maximal norm at least $c\log\log N$ in every ordering, matching Bourgain's upper bound.

We prove that $L^2(\Omega)$ admits no Riesz basis of exponentials when $\Omega$ is a ball in $\R^n$, $n\geq2$, or a nondegenerate triangle in $\R^2$. The ball result extends to every nonempty bounded convex open set with $C^2$ boundary in $\R^n$, $n\geq2$. Starting from a hypothetical Riesz basis, we use Beurling weak limits of translates of its frequency set to construct a stationary system. On the associated Hilbert space, we define spectral cutoff projections and a comparison family of orthogonal projections obtained from the synthesis operator. The distance between corresponding projections is uniformly bounded by a constant strictly smaller than one. The comparison family is continuous in operator norm, whereas the boundary geometry of $\Omega$ produces two strong limits of the cutoff projections with strictly nested ranges. Both limiting projections would then be at distance less than one from the same comparison projection, which is impossible.

We investigate the optimal endpoint $L^2$ restriction estimates of Laplace eigenfunctions on submanifolds of codimension 2 in a smooth closed Riemannian manifold $M$. For every fixed smooth codimension-two submanifold $\Sigma$, we prove a little-o improvement $o(\lambda^{1/2}\sqrt{\log\lambda})$ on the classical estimate $O(\lambda^{1/2}\sqrt{\log\lambda})$ of Burq–Gérard–Tzvetkov and Hu. Our proof uses the Bargmann transform and Tataru's phase-space representation to reduce the problem to Stein–Street's estimate for singular Radon transforms. The three-dimensional case can be handled directly by Ricci–Stein's estimate. Moreover, we construct explicit examples to show that the little-o improvement is optimal in general. These are closely related to earlier counterexamples for endpoint Strichartz estimates by Montgomery–Smith and Carbery–Hofmann. In particular, we establish the log-free estimate $O(\lambda^{1/2})$ when $(M,g)$ and $\Sigma$ are real analytic, whereas this estimate fails in general in the smooth setting.

The Ramanujan$-$$q$-Airy connection formula relates the convergent series at the origin to the behaviour at infinity of the Ramanujan entire function. We extend this connection to a one-parameter deformation, which embeds the Ramanujan (second-order) $q$-difference operator in a family of third-order equations. By contour integral as $q$-Borel inversion, we give behaviour at infinity in terms of divergent local expansions with connection coefficients uniquely determined. Discrete $q$-Borel$-$Laplace summation yields a convergent, bilateral power series representation, whose dependence on summation path reflects the $q$-Stokes phenomenon. We prove remainder estimates establishing the formal, generally divergent, expansion at infinity as an asymptotic description of the entire function.

Let $z_{n,j}(\lambda)$ denote the positive zeros, in decreasing order, of the ultraspherical polynomial $C_n^\lambda$, $\lambda>-1/2$, with the reduced limiting interpretation at $\lambda=0$ specified below. Our principal result settles three higher-monotonicity questions of Gautschi: two as printed and the natural open-interval form of the third, whose printed endpoint $\lambda=0$ is singular. For every $n\geq3$, $$ \sqrt{\lambda+1}\,z_{n,j}(\lambda) $$ becomes a complete Bernstein function after translation of its parameter interval to $(0,\infty)$. For every $n\geq2$, $$ \sqrt{\lambda}\,z_{n,j}(\lambda) $$ is a complete Bernstein function on $(0,\infty)$. For every $n\geq4$, the largest-zero trajectory $$ \sqrt{\lambda+\frac{2n^2+1}{4n+2}}\,z_{n,1}(\lambda) $$ has the same property, whilst for every other positive zero the derivative of this scaling fails to be completely monotone. Separately, an exact calculation in degree $4$ provides a counterexample to a fourth conjecture of Gautschi, concerning the linear scaling $\lambda z_{n,j}(\lambda)$.

We introduce several notions of resonance for systems exhibiting weak or partial dissipation: $\dot u=Au+f(t)$, with $A$ the generator of a strongly stable semigroup $S(t)$ on a Hilbert space $H$. We refer to resonance as the phenomenon where a time-periodic $f$ may yield an unbounded $u$, classically occurring when $A$ has imaginary eigenvalues. In infinite dimensions, however, unbounded growth may depend on topological choices, and several “resonances" may occur. Classically, existence of $T$-periodic $u$ under $T$-periodic $f$ is equivalent to a range condition: $\mathcal{R}(I-S(T))=H$. Consequently, its failure permits unbounded solution growth and we elucidate that connection through properties of $S(t)$. We describe a hierarchy of range failures, and provide an associated resonance taxonomy. The growth rate of ~$||(A-\lambda I)^{-1}||_H$ on ~$i\R$ dictates a regularity gap between $u$ and $f$, and, for rapid growth in $\lambda$, that gap can be infinite. In this case, we demonstrate implications for periodic solvability, and a mechanism for constructing smooth resonant forces. The theory developed here is motivated by (and demonstrated for) a hyperbolic-parabolic system, where the latter component provides the only system dissipation. Such dynamics are a simplification of fluid-structure phenomena, which demonstrate strong stability but do not provide unconditional periodic well-posedness. Though point-spectral resonance is ruled out, we note that $\mathcal R(I-S(T))\neq H$. Our main result here shows that, for a carefully constructed geometry, a classical heat-wave system possesses periods $T$ which yield infinite derivative loss and, via supporting results, yield resonance with smooth forcing.

Let $n \geq 5$. We prove that the helical maximal operator is bounded on $L^p(\mathbb{R}^n)$ for some $p<2n-4$. This improves the previously known range $p>2n-4$ obtained by Gan-Maldague-Oh. As observed by Beltran-Hickman, the exponent $2n-4$ is the natural limit of the standard local smoothing method.

Let $\Th(\alpha,L)=\sum_{v\in L}e^{-\pi\alpha|v|^2}$ for $\alpha>0$ and $E(L,s)=\sum_{v\in L\setminus\{0\}}|v|^{-2s}$ for $s>3/2$ be the theta and Epstein zeta functions associated to the lattice $L$, respectively. We are particularly interested in physically relevant dimension three. Fix the covolume of the lattice $L$ to $1$. Up to an orthogonal transformation, we prove that \begin{equation}\nonumber \argmin_{|L|=1}\Th(\alpha,L)= \begin{cases} \boldsymbol{\mathrm{FCC}}\;\;\mathrm{lattice},\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;&\text{if}\;\; \alpha>1, \boldsymbol{\mathrm{BCC}}\;\;\mathrm{lattice}, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&\text{if}\;\; \alpha<1, \boldsymbol{\mathrm{FCC}}\;\;\mathrm{or}\;\; \boldsymbol{\mathrm{BCC}}\;\;\mathrm{lattice},\;\;&\text{if}\;\;\alpha=1, \end{cases} \end{equation} and \begin{equation}\nonumber \argmin_{|L|=1}E(L,s)= \boldsymbol{\mathrm{FCC}}\;\;\mathrm{lattice},\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\text{if}\;\; s>3/2. \end{equation} Therefore, we prove the Sarnak–Strömbergsson conjecture \cite[Inequalities (43)–(44), Section 5]{SS}.

We construct singular weak solutions to scalar, linear, uniformly parabolic equations with complex coefficients in one space dimension. Our examples show that the space-time integrability provided by the energy estimates is sharp: for every $p>6$, there exists such an equation admitting an energy solution that fails to be $p$-integrable on a compact subset of the space-time domain. In particular, the local boundedness of weak solutions from the De Giorgi–Nash–Moser theory fails for complex coefficients already in one space dimension.

We improve the exponents of the $A_p$ constant in the strong and weak-type weighted bounds for the strong maximal function, for every $1 < p <\infty$ and every dimension $d \geq 2$. In dimension two, the strong and weak $L^2(w)$ exponents improve from $2$ to $7/4$ and from $3/2$ to $5/4$, respectively. The proof exploits the geometry of pairwise intersections of antichains of dyadic rectangles.

Let $U,V$ be matrix $\mathcal A_p$ weights, $1<p<\infty$, and let $B$ be a matrix-valued function. For Calderón–Zygmund operators $T$ with scalar standard kernels, we characterize boundedness and compactness of $[M_B,T\otimes I_m]:L^p(U)\to L^p(V)$ by two-matrix BMO and vanishing oscillation conditions. The converse statements require non-degeneracy condition. For matrix \(\mathcal A_2\) weights \(U,V\) and a fixed Riesz transform \(R_j\) on \(\mathbb R^d\), \(d\geq2\), we characterize the Schatten class membership of \(C_{B,R_j}:L^2(U)\to L^2(V)\). We characterize $\mathcal S^r$ membership for $r>d$ and $\mathcal S^{d,\infty}$ membership by $\ell^r$ and weak $\ell^d$ conditions on the two Hilbert–Schmidt cube oscillations. For $0<r\leq d$, $\mathcal S^r$ membership is equivalent to $B$ being constant almost everywhere. The weak endpoint condition implies a weighted first-order Sobolev estimate for $C^1$ symbols, with a converse for constant or uniformly elliptic weights.

We determine exactly how harmonic sums, taken over integers with a given number of occurrences of a fixed block of digits, depend on that number of occurrences. Language factorizations, combined with stochastic radix expansions, relate these harmonic sums to the iteration of an operator whose eigenvectors are the Lebesgue measure and distributional derivatives of singular measures. The dual picture is given by a Riesz basis of eigenpolynomials in a suitable Hardy space. The resulting modal expansion admits a natural interpretation as a block-directed Euler-Maclaurin formula interpolating between Euler-Maclaurin and Taylor expansions.

We prove that sets in a certain class do not admit Riesz bases of exponentials. In particular, this class contains disks and triangles in the plane.

The recent work \cite{HeWuXiaZhang2025} introduced a finite-data inverse nodal framework for regular one-dimensional Sturm-Liouville operators. Nevertheless, formulating a multidimensional inverse nodal theory for Schrödinger operators remains a challenging open problem. This paper addresses this gap by establishing a finite radial inverse nodal theory for the radial spectral branch of Schrödinger operators in balls. In contrast with the interval case, the nodal data are the radii of spherical nodal hypersurfaces of radial eigenfunctions. The exact radial reduction preserves this PDE nodal geometry and leads naturally to a singular weighted Sturm–Liouville problem with weight $w(r)=r^{n-1}$, rather than to a regular interval model. We minimize the PDE distance $\|Q-Q_0\|_{L^p(B_R)}$ , equivalently the weighted distance $\|q-q_0\|_{L^p_w(0,R)}$, among all radial potentials matching finitely many prescribed spherical nodal radii. For the associated nodal constraint sets, we prove spectral regularity, nodal differentiability, weak closedness, exact realization, and a finite-codimensional $C^1$ manifold structure. Crucially, without the variational selection imposed by \(q_0\), the prescribed finite nodal radii fail to determine the potential uniquely: the admissible potentials are invariant under constant shifts, and the finite nodal constraints yield locally infinite-dimensional level sets at regular points.

In this short note, we show that the loss of order $N^{c_p/\log\log N}$ in the version of the Ionescu–Wainger multiplier theorem for canonical fractions, recently established by Kosz, Mirek, Słomian, and Zhang in arXiv:2606.12663, is optimal.

In their previous work devoted to the distribution of rational points near Brownian motion, the authors conjectured the existence of an oscillation principle governing the asymptotic behavior of the number of rational points with bounded denomi\-nators near the graph of a monofractal curve. In this note, a weaker form of this conjecture is shown to hold for a broad class of deterministic fractal curves. These include the classical Takagi and Weierstrass nowhere differentiable functions, and indeed a prevalent (i.e.~"large") class of functions among those which are Hölder continuous with a given exponent of regularity. This constitutes the first instance of deterministic fractal curves for which a precise count of the rational points under consideration is established.

Nested cubature rules, in which each refinement retains all previously used nodes and thus reuses earlier function evaluations, are natural in multilevel and adaptive integration. For every fixed $s>d/2$, we show that the equal-weight QMC integration rate on $\mathbb S^d$ is compatible with such nested point sets. In the subcritical range $d/2<s<d$, cumulative unions of geometrically growing QMC blocks yield nested QMC design sequences whose successive cardinality ratios converge to any prescribed $\rho>1$. At and above the critical index $s=d$, where the block-averaging estimate no longer yields the optimal rate, we prove an equal-weight completion theorem based on low-frequency discrepancy cancellation. A block-sensitive estimate sharpens the iteration and yields nested QMC designs for all $s\ge d$ with $N_{j+1}\lesssim(j+1)^{2s/d-1}N_j$. Every prescribed finite point set also admits an optimal-rate completion.

The commutators of intrinsic square functions associated with the intrinsic Littlewood–Paley $g$-function, the intrinsic $g_{\lambda}^{*}$-function and the intrinsic Lusin area function can be used to characterize BMO-type function spaces through their boundedness properties. In this paper, we show that, for $b\in {\rm BMO}(\mathbb{R}^{n})$, the boundedness of each of these commutators from the Musielak–Orlicz Hardy space $H^{\varphi}(\mathbb{R}^{n})$ to $L^{\varphi}(\mathbb{R}^{n})$ is equivalent to $b\in {\rm BMO}_{\varphi}(\mathbb{R}^{n})$ (a nontrivial subspace of $\rm{BMO}(\mathbb{R}^{n})$), under suitable assumptions on the growth function $\varphi$. This extends the weighted characterization of Han and Wu [Proc. Amer. Math. Soc. 152(1) (2024), 281–293] to the Musielak–Orlicz setting. In addition, under suitable assumptions on a growth function $\Phi$, we prove that, for $b\in {\rm BMO}_{\Phi}(\mathbb{R}^{n})$, these commutators are bounded from the $b$-adapted Orlicz–Hardy space $H_{b}^{\Phi}(\mathbb{R}^{n})$ to $L^{1}(\mathbb{R}^{n})$ and from Orlicz Hardy space $H^{\Phi}(\mathbb{R}^{n})$ to $L^{1,\infty}(\mathbb{R}^{n})$.

We prove that Besicovitch's covering theorem holds for the parabolic metric $d_p((x_1,t_1),(x_2,t_2))=(|x_1-x_2|^{p}+|t_1-t_2|^{\frac{p}{2}})^{\frac{1}{p}}$ on $\mathbb{R}^n\times\mathbb{R}$ if and only if $p\geqslant 2$. If $p=2$, this answers a question posed by P. Mattila in the affirmative.

The spherical maximal operator of order $\mu$ was introduced by El Kohen in the setting of real hyperbolic spaces, where its $L^p$-boundedness properties were studied. In this paper, we extend this notion to rank-one Riemannian symmetric spaces of noncompact type. We establish sufficient conditions on the parameter $\mu$ for the corresponding spherical maximal operator to be bounded on $L^p$ for $1<p\leq\infty$. We also obtain a necessary condition for $L^p$-boundedness when $\mu$ is real and $1<p<\infty$. In particular, for $1<p\leq2$, the necessary condition agrees with the sufficient condition, yielding the sharp range of real $\mu$ in this regime.

Let $R_n^{(a,b)}=P_n^{(a,b)}/P_n^{(a,b)}(1)$ be the normalised Jacobi polynomials. For $\alpha\ge0$, put $\xi_0=-1$ and let $\xi_n$ be the largest zero of $R_n^{(\alpha+1,\alpha)}$ for $n\ge1$. We prove that $$ R_k^{(\alpha,\alpha)}(x) =\min_{n\ge0}R_n^{(\alpha,\alpha)}(x), \quad \xi_{k-1}\le x\le\xi_k, $$ for every $k\ge1$. This determines the pointwise lower envelope of the normalised ultraspherical polynomials and answers Question 2 posed by F. M. de Oliveira Filho in his 2009 thesis. For $\alpha>0$ and $-1<x<1$, we also determine all minimising degrees. The sequence is non-increasing up to each such degree, which answers his Question 1 in this range. A Legendre example gives a negative answer when $\alpha=0$.