Papers

We embed the Jacobi-like and Laguerre-like systems for ordinary type-I/type-II multiple orthogonality introduced by Wolfs into a single Askey-type confluence scheme. Applying the Bernstein transform simultaneously to the $q$ Jacobi-like weights produces a positive finite-lattice Hahn-like ancestor, represented by multiple beta integrals, reducing to the classical Hahn weight for $q=1$, and converging back to the continuous system. Parameter and scaling limits then yield Kravchuk-like, two Meixner-like, two Charlier-like, Jacobi-like, two Laguerre-like, and Hermite-like families. We realize every arrow through explicit hypergeometric orthogonality data. For each near-diagonal multi-index $1\leq |\mathbf m|\leq N$, the Hahn-like type-II polynomial is a terminating ${}_{q+2}F_{q+1}(1)$ series satisfying an exact inverse Bernstein identity. We construct the normalized type-I form and, under explicit separation and nonvanishing assumptions, recover its individual polynomial components by finite sums of terminating hypergeometric functions and prove uniqueness. Finite-pole terms regroup into terminating Kampé de Fériet blocks. Their limits are obtained by coefficient extraction and reconstruction at infinity on the unreflected Kravchuk-like and Meixner-II-like branches, and by finite Lauricella–Horn sector sums on the reflected branches. This yields sectorwise component confluence along the Kravchuk–Charlier, Meixner–Charlier, Meixner–Laguerre, and Charlier–Hermite arrows. For $q=1$ all formulas reduce to the classical families; for $q>1$ the two Meixner-like systems, and likewise the two Charlier-like systems, are inequivalent under permutation of normalized rows.

The solvability of two-point boundary value problems for constant-coefficient differential-algebraic equations is investigated. Unlike previous studies, which often assume that the matrix pair associated with the equation is regular, we consider the general singular case. Using the Kronecker canonical form, we decompose the problem into simpler subsystems, enabling a systematic analysis of solvability. The method of parameterization reduces the boundary value problem to a system of algebraic equations. We derive criteria for the existence and uniqueness of solutions and provide a comprehensive framework for solving such boundary value problems. Several illustrative examples are presented and show the applicability of the results.

For every finite family of Euclidean balls in $\mathbb{R}^n$, $n\ge2$, and every $0<\delta<1/2$, we select a subfamily whose $(1+\delta)$-dilations cover the original union and whose undilated balls have multiplicity at most $C_n\delta^{-(n-1)/2}$. This proves the covering estimate conjectured by Han and Lu \cite{HL}. As an application, we determine the optimal Solyanik asymptotic \[ \mathcal C_n(\alpha)-1\asymp_n(1-\alpha)^{2/(n+1)} \qquad(\alpha\uparrow1) \] for the uncentered Hardy–Littlewood maximal operator over Euclidean balls, which is Conjecture~1(b) of Hagelstein and Parissis \cite{HP14}. For the modified uncentered maximal operators, we determine the optimal weak $(1,1)$ growth rate $(k-1)^{-(n-1)/2}$ as $k\downarrow1$, uniformly over Radon measures. We also obtain the corresponding $L^p$ and Fefferman–Stein bounds.

Let $G_{d}$ be a $d \times d$ matrix with independent standard complex Gaussian entries, let $\alpha_{\mathbb{C}}(d)$ be the expected average singular value of $G_{d}/\sqrt{d}$, and set $\Delta_d := \alpha_{\mathbb{C}}(d)-\alpha_{\mathbb{C}}(d+1)$. The statistic $\alpha_{\mathbb{C}}(d)$ admits a variational representation as an expected normalized maximum over the unitary group and governs approximation guarantees for the little Grothendieck problem over the unitary group and related unitary registration problems. We obtain a strictly positive lower bound and an upper bound for $\Delta_d$, both valid in every dimension, together with corresponding bounds for $\alpha_{\mathbb{C}}(d)$ around the Marchenko–Pastur limit. These bounds match the sharp leading behavior of complete asymptotic expansions for both quantities, whose coefficients are explicitly computable. The proof combines a three-term recurrence for $Y_d = d^{3/2}\alpha_{\mathbb{C}}(d)$, obtained from its continuous dual Hahn representation, along with singularity analysis of the underlying Laguerre moment generating function.

Fix an integer $K\ge2$, and let $C_K^n =\{(e^{\frac{2\pi ij}{K}})_{j=0}^{K-1}\}^n$ be the product of cyclic groups of order $K$. For a Fourier character $\chi_\alpha$, let $s(\alpha)$ be the number of active coordinates. We give a self-contained proposed proof that the dimension-free Bohnenblust–Hille constants governed by interaction order grow polynomially: if $p_d=2d/(d+1)$ and \[ \gamma_2=\frac12, \qquad \gamma_K=\frac{K\log(K-1)}{4(K-2)}\quad(K\ge3), \] then the $\ell^{p_d}$ norm of Fourier coefficients $\{\hat f(\alpha)\}$ is bounded by $L^\infty$ norm of $f$ multiplied by $C(K) d^{4\gamma_K+5}$. The constant $C(K)$ is actually at most of the order $K^{5/2}$.

In this note, we give a counterexample to the assertion of Problem 10.4 in Bugeaud's monograph Distribution modulo one and Diophantine approximation, which goes back to Mendès France. The problem states that the spectrum of the sequence $(\xi\alpha^n)_{n\ge1}$, that is, the set of irrational $\theta\in(0,1)$ for which $(\xi\alpha^n-n\theta)_{n\ge1}$ is not uniformly distributed modulo one, is at most countable for all real $\xi\ne0$ and $\alpha>1$. More precisely, we prove that for every real $\alpha>1$ there are $2^{\aleph_0}$ real numbers $\xi>0$ for which the spectrum of $(\xi\alpha^n)_{n\ge1}$ contains one and the same uncountable set.

We study mixed multiple orthogonal polynomials associated with rank-one matrices of measures and their recurrence relations. On the step line we identify the corresponding banded recurrence matrix through the Gauss–Borel factorization of the moment matrix. Elementary Christoffel transformations lead to a factorization of this recurrence matrix into lower and upper bidiagonal factors. For the mixed Piñeiro system we derive contour-integral and generalized hypergeometric representations and formulas for the recurrence and factorization coefficients.

We show the following equivalence of solutions to general fractional Helmholtz equations: $u$ is a solution to $$A^su=\lambda^su$$ for some fixed $0<s<2$ and some $\lambda>0$ if and only if $u$ is a solution to the same equation for all $s\neq0$. Furthermore, we show as well that $u$ equivalently solves the logarithmic Helmholtz equation $$\log(A)u=(\log\lambda)u.$$ Here, $A$ is a nonnegative, linear operator on a Banach space $X$. In particular, our results hold whenever $A=-L$, where $L$ is the infinitesimal generator of a $C_0$-semigroup in a Banach space $X$ satisfying mild assumptions that are typical in applications. As a particular case, we recover known results for the fractional Laplacian $(-\Delta)^s$ in $\mathbb{R}^n$. More importantly, we provide a list of examples of other fractional power operators in settings where the Fourier transform is not available and for which our theorems apply. These include fractional powers and logarithms of second order elliptic operators in bounded domains, Laplace–Beltrami operators on Riemannian manifolds, space-time master equations, the discrete Laplacian, and fractional derivatives.

We prove the boundedness of multilinear Hilbert transforms along moment curves by establishing uniform estimates for both translated and untranslated multilinear paraproducts and by applying a Sobolev smoothing inequality. As a byproduct of our method, we obtain the same bounds for multilinear maximal functions along moment curves.

One-parameter variational inequalities are well developed, whereas their multi-parameter counterparts remain much less understood. We explore two-parameter variational inequalities for averages over tori in $\mathbb{R}^3$. To capture the underlying two-parameter structure, we introduce a local two-parameter $r$-variation norm that combines rectangular increments with variations along the boundary. The resulting variation operator pointwise dominates the corresponding two-parameter local maximal function and, unlike the maximal function, also captures oscillation across the two parameters. We establish sharp $L^p$–$L^q$ bounds for this variation operator up to endpoints. For comparison, we also obtain sharp $L^p$ bounds up to endpoints for the corresponding local one-parameter variation operator, revealing a genuine difference between the one- and two-parameter boundedness regions. The proof combines square function estimates for two-parameter propagators with local smoothing estimates through mixed-norm interpolation.

Robustness of exponential dichotomies against small-delay perturbations presents a fundamental obstacle: the lack of an effective admissible characterization in nonautonomous delay equations. Considerable efforts have been devoted to obtaining such a characterization for differential equations in Banach spaces. However, even unlike ordinary differential equations, the variation of constants formula for delay equations requires extending the phase space to a space of discontinuous functions, and the dependence on the past states leads to another challenge in establishing exponential dichotomy via admissibility, i.e., constructing a suitable admissible pair and then estimating the growth/decay rates along the unstable/stable subspaces via the admissibility property. In this paper, we provide an admissible characterization based on two pairs of Banach spaces that yields explicit dichotomy exponents. Through this characterization and an operator perturbation method, we prove that small-delay perturbations preserve exponential dichotomies.

In this paper, we consider Kakeya sets with the additional restriction that centers of the unit line segments belong to a given set. In particular, for every uncountable Borel set $A\subset\mathbb{R}^d$, $d\geq 2$, we construct a compact subset of $\mathbb{R}^d$ of Lebesgue measure zero that contains, in every direction, a unit line segment whose center lies in $A$. Notice that every such Kakeya set (not even necessarily compact) must have positive Lebesgue measure if $A$ is countable. So our result shows that the countability is in fact the only obstruction.

The Donoho–Stark uncertainty principle asserts $N^d\le|E|\,|\Sigma|$, and the additive energy uncertainty principle of Aldahleh–Iosevich–Iosevich–Jaimangal–Mayeli–Pack refines it to $1\le|E|\cdot\|1_\Sigma\|_{U^2}^{4/3}$. It has been conjectured that the natural higher-order analogue \[ 1\le |E|\cdot\|1_\Sigma\|_{U^k}^{2^k/(k+1)} \] holds for every $k\ge 2$, the coset case giving equality. We show that this fails for every $k\ge 6$. The counterexample is the two-point signal $f=\delta_0-\delta_1$ on $\mathbb{Z}_3$, for which the left-hand quantity equals $2/3(2(k+1))^{1/(k+1)}$; this is $<1$ precisely when $k\ge 6$, and tends to $2/3$.

We prove that many curved Kakeya problems, originally motivated by Hörmander's oscillatory integral problem, lift to the classical Kakeya problem in higher dimensions. As a consequence, if the classical Kakeya set conjecture were true in all dimensions, the families of curves whose curved Kakeya sets have full dimension are dense among Hörmander-type families. This is in contrast to the nowhere dense Bourgain's condition, which is a necessary condition for best-case curved Kakeya maximal function estimates. Furthermore the linear Kakeya set conjecture would imply a fairly complete understanding of Kakeya sets of quadratic curves, as first studied systematically by Wisewell. These results give a better understanding of a question posed by Guo–Guth–Nadjimzadah–Shen–Zhang, and they more generally show that the classical Kakeya conjecture in high dimensions rests on a broad range of curved Kakeya problems in lower dimensions.

We prove the exact asymptotics of the multidimensional normalized $L^1$ Nikolskii constant \[ \mathcal L^*(d)=\Bigl(\frac{\pi}{2}+o(1)\Bigr)2^{-d}, \quad d\to\infty. \] In addition, for each fixed dimension we obtain asymptotics for the positive zeros of the extremal function $\varphi_d$, and determine their limiting distribution as $d\to\infty$. The lower bound follows from the construction of an admissible function and its asymptotic analysis. For the upper bound, we represent $x^{d+1}\varphi_d(x)$ as the product of two solutions of the equation $u''+V_d(x)u=0$, compare this equation with a Bessel model, and analyze relative canonical products.

There is a close relationship between divided differences and hypergeometric series, and recent studies have shown that divided differences can be used effectively to derive terminating hypergeometric identities. In this paper, we apply this approach to terminating zero-balanced hypergeometric series. Using explicit product evaluations arising from divided differences, we give alternative proofs of zero-balanced ${}_3F_2(1)$ and ${}_4F_3(1)$ summation formulas and then extend the argument to the general terminating ${}_{r+1}F_r(1)$ case. We also combine the Lagrange representation with the Leibniz rule for divided differences to derive a finite convolution transformation for a terminating ${}_{r+2}F_{r+1}(1)$ series. Thus, the divided-difference approach not only reproduces the known zero-balanced summation formulas but also yields a finite convolution transformation for terminating hypergeometric series, together with its zero-balanced specialization.

For a compact set $K\subset\mathbb{C}$, let $\vartheta(K)$ be the infimum of the planar areas of the unit lemniscates of all monic polynomials with zeros in $K$, allowing arbitrary degree and repeated zeros. We prove that $\vartheta(K)=0$ whenever $\operatorname{cap}(K)=1$, with no regularity assumption on $K$. The proof uses a centered harmonic polynomial that is positive on all but a set of arbitrarily small area in the polynomial hull of $K$. A Fourier average of exterior harmonic measures realizes this polynomial as the logarithmic potential of a signed measure having bounded density with respect to the equilibrium measure. A positive perturbation and an $L^1$ approximation by empirical measures then produce the required polynomials. This extends the smooth-boundary result of Krishnapur, Lundberg, and Ramachandran to arbitrary compact sets of capacity one. Together with the capacity-greater-than-one theorem of Ghosh and Ramachandran and an elementary argument for unbounded sets, it follows that $\vartheta(F)=0$ for every closed infinite set $F\subset\mathbb{C}$ of transfinite diameter at least one, answering the vanishing question in Erdős Problem 1040.

We prove that the gradient range of every nonzero compactly supported $C^1$ function on $\mathbb{R}^n$ is the closure of its interior, for every $n\geq1$. More generally, for an arbitrary $C^1$ function, the closure of its gradient range is the union of the closure of the range's interior and the set of asymptotic gradient values. We also prove that compact smooth Lagrangian submanifolds without boundary in $\mathbb{R}^{2n}$ and their images under symplectic homeomorphisms have regular closed momentum projections.

A discrete set $\Lambda$ is called a spectrum of a Borel probability measure $\mu$ if the exponential functions $\{e^{2\pi i \langle\lambda, x\rangle}:\lambda\in\Lambda\}$ form an orthonormal basis for $L^2(\mu)$. In this work, we first give necessary and sufficient conditions for a self-replicating translation set to be a spectrum of the self-similar measure associated with a product-form Hadamard triple, which were discovered by the first-named author and Lai [Adv. Math. 431 (2023) Paper No.109257]. These results extend the studies by Łaba and Wang [J. Funct. Anal. 193 (2002) 409-420], Dutkay and Lai [J. Math. Pures Appl. 107 (2017) 183-204] in $\mathbb R$. As an application, we can explicitly construct such a self-replicating spectrum. Finally, we prove that the dual spectral set conjecture holds for a self-replicating translation set.

Fix a base $b\ge2$ and a digit set $\mathcal{D}\subset\{0,1,\dots,b-1\}$, and let $\mathcal C_k$ be the set of integers in $[0,b^k)$ whose $k$ base-$b$ digits all lie in $\mathcal D$. We study prime pairs and three-term arithmetic progressions in primes whose difference, or common difference, belongs to $\mathcal C_k$. Under a natural digit non-resonance condition, we establish a localized Fourier criterion guaranteeing asymptotic formulae with explicit leading constants for both problems. Using a transfer-operator argument, we verify this criterion in explicit parameter ranges for several natural families of digit sets.

In this paper we state, prove, and discuss some new boundedness results concerning sequences of linear functionals of functions from the Lipschitz class \(Lip_1\) and general orthonormal systems. It is also proved that these results are, in a sense, the best possible.

Let \(T_\Lambda=\mathbb R^d/\Lambda\), \(d\geq2\), be a flat torus with quotient metric \(\rho_\Lambda\), and consider the distance energies \(I_\alpha(\mu)=\iint \rho_\Lambda(x,y)^\alpha\,d\mu(x)\,d\mu(y)\) of Borel probability measures \(\mu\). We prove a quantitative Fourier signature of the cut locus: for every Voronoi facet and every \(\alpha>0\), there is a sequence of dual-lattice frequencies approaching the facet normal along which the Fourier coefficients of \(\rho_\Lambda^\alpha\) are positive, with an explicit leading asymptotic determined by the facet. As direct consequences, Haar measure is not a local maximizer for any positive distance power, even among smooth densities, and every flat torus of dimension at least two has supremal negative type and generalized roundness zero. We then solve the global maximization problem for two classes of flat tori. On an orthogonal rectangular torus, the maximizers undergo a transition at \(\alpha=2\): for \(1\leq\alpha<2\) they are the translated uniform measures on the two-torsion subgroup; at \(\alpha=2\) all balanced couplings on translates of that subgroup are extremal; and for \(\alpha>2\) only equally weighted diametral pairs remain. On the regular hexagonal torus, the maximizers are precisely the uniform measures on translates of a distinguished cyclic subgroup of order three for every \(\alpha\geq1\).

We prove a weak-type $(1,1)$ estimate for the centered Hardy–Littlewood maximal function with respect to Euclidean balls with dimensional dependence $O(\sqrt{n} \log n)$. This improves the order of growth in the classical $O(n)$ estimate of Stein and Strömberg. The proof goes through a pointwise bound of the Hardy–Littlewood maximal operator by the heat maximal operator with $\sqrt{n}$ loss. The key technical aspect of our result is an improvement of the weak-type bound for the heat maximal operator from $O(\sqrt{n})$ to $O(\log n)$.

For a finite set $S$ of positive integers, put $K(S):=-\min_{x\in\mathbb T}\sum_{s\in S}\cos(2\pi sx)$. Bedert recently proved the uniform lower bound $K(S)\geq |S|^{1/5-o(1)}$. We remove the subpolynomial loss and prove that $K(S)\geq c|S|^{1/5}$ for an absolute constant $c>0$. The proof combines two estimates from Bedert's argument with an exact averaging identity for the asymmetric boundaries of additive intersections. This identity replaces the multiplicative-amplification step responsible for the logarithmic loss.

We compute the limit of the eigenvalue of the $k$-Hessian operator on the unit ball in $\mathbb{R}^n$ when $k\rightarrow \infty$, assuming that the ratio $\frac{n}{k}$ remains fixed. When $n< 2ke$, we moreover identify the limit of the corresponding eigenfunctions. We also derive monotonicity results and consider when the ratio varies in $n$.

We investigate the spectral properties of a class of inhomogeneous self-similar measures, which does not admit a non-trivial infinite convolution structure. A central example is the golden-mean self-similar measure $\mu$, for which the existence of an exponential orthonormal basis in the associated $L^2$-space has remained a long-standing open problem. The usual approach for homogeneous self-similar measures does not apply here, new methods are required. We establish several basic properties of the measure and then carry out a detailed numerical study of the zero set of its Fourier transform. Using a scanning and refinement algorithm that combines uniform grid sampling, quadratic interpolation, and golden-section search, we examine a wide range and find no real zeros of $\widehat{\mu}$, which provides concrete evidence that $\mu$ is very likely non-spectral, suggesting that inhomogeneity may serve as a natural obstruction to the existence of exponential orthonormal bases. To the best of our knowledge, our paper is the first attempt to study the spectrality of such measures through a combined analytic and numerical framework.

We study the complete number problem for a class of self-similar spectral measures on the real line. For a spectral pair $(\mu,\Lambda)$, a real number $t$ is called complete if $t\Lambda$ is also a spectrum of $\mu$. In this paper we consider the $N$-Bernoulli convolution $\mu_{N^r,\mathcal D},\mathcal D=\{0,1,\ldots,N-1\},$ together with its spectrum $\Lambda_{N^r,N^{r-1}\mathcal D}.$ Our main result establishes that every prime number, apart from certain trivial cases, is complete.

We present an exposition of Bourgain's celebrated 1986 proof of the circular maximal theorem.

We study the near $L^1$ behavior of the maximally quadratically modulated Hilbert transform \[ \mathcal{C}_2f(x) := \sup_{\lambda} \left|\operatorname{p.v.} \int_{\mathbb{R}}f(x-y)e^{2 \pi i \lambda y^2} \frac{\mathrm{d} y }{y} \right| \] and its lacunary counterpart obtained by restricting $\lambda$ to $2^{\mathbb{Z}}.$ We prove that if $\Phi$ is a Young function satisfying $ \Phi(t)=o\bigl(t\log_2t\bigr)$ then $\mathcal{C}_{2,\mathsf{lac}}$, and therefore $\mathcal{C}_2$ as well, does not satisfy a corresponding $\Phi$-modular estimate. In particular, neither the lacunary nor the full quadratic operator is of weak type $(1,1)$. In the positive direction, we establish an $L\log_1 L$ modular estimate for $\mathcal{C}_2$ and a $L (\log_2L)^2 \log_4L$ estimate for $\mathcal{C}_{2,\mathsf{lac}}.$

We prove that the electric field generated in three-dimensional space by finitely many point charges of one sign has only finitely many equilibrium points, thereby answering a 1969 question of Morse and Cairns (restated by Eremenko in 2008 and, as a conjecture, by Shapiro in 2015). More generally, for nonzero charges $q_i$ of possibly mixed signs at distinct sites $\mathbf a_i\in\mathbf{R}^3$, we show that the Coulomb field has at most $2^{N-4}(N-1)(9N^2+9N+10)$ equilibria in the region where $S(\mathbf x):=\sum_iq_i|\mathbf x-\mathbf a_i|^{-3}$ does not vanish. Of course, for charges of one sign, $S$ is nonzero everywhere. The proof rules out curves of equilibria using algebraic geometry and complex analysis on an associated complex curve, and then applies a Bézout count to obtain a quantitative bound. Well-known examples show that, in the mixed-sign case, the Coulomb field can vanish on curves contained in the zero set of $S$.