For every $0\leq\beta\leq1/2$, we construct a nonzero real-valued continuous function $f_\beta$ in $L^1(\mathbb R)\cap L^2(\mathbb R)$ such that $\widehat {f}_\beta=f_\beta$ and $f_\beta(\sqrt{n}/[\log(e+n)]^{\beta})=0$ for all $n\geq 0$. The case $\beta=0$ settles in the negative a question raised by Radchenko and Viazovska regarding their Fourier interpolation formula. The construction uses a scale of reproducing kernel Hilbert spaces generated by the Fourier-invariant Hermite functions. Applying the Mehler formula, we identify the reproducing kernels of these spaces. By suitable estimates of these kernels, we show that $(\sqrt{n}/[\log(e+n)]^{\beta})$, with one auxiliary point added to it, is a universal interpolating sequence for at least one of the Hilbert spaces under consideration. However, this result fails when $\beta>1/2$.
Let $(L,e^{-\phi})$ be a positive Hermitian holomorphic line bundle over a compact Riemann surface $X$, and let $\omega=i\partial\overline\partial\phi$. We obtain explicit pointwise estimates for the Bergman form of the tensor power $mL$. If $\mathrm{Ric}\omega\leq\omega$ and the shortest nonconstant closed geodesic has length at least $2\pi$, then \[ K_{m\phi}\geq \frac{2m-1}{4\pi}\,\omega, \] with sharpness holding for $(\mathbb P^1,\mathcal O_{\mathbb P^1}(2))$. We also obtain a local version, depending on an upper curvature bound and the injectivity radius, which recovers the first two terms of the Bergman expansion when the curvature is constant. Under the two-sided bound $-\omega\leq\mathrm{Ric}\omega\leq\omega$ and the same closed-geodesic hypothesis, we also prove \[ K_{m\phi}\leq \frac{m\omega}{2\pi} \left(1+\frac{54.8\log(2m)}{m-\frac{1}2}\right). \] The lower estimates use the deformation to the tangent space version of the Ohsawa--Takegoshi theorem established by He, Wang, and the author, whereas the upper bound is obtained by observing the submean inequality with quantitative isothermal coordinates that were obtained in recent work by Eilat.
Let $\mathbb F_2$ be the finite field with two elements and let $\mathbb F_2^\omega$ denote the countably infinite-dimensional vector space over $\mathbb F_2$. We show that, unlike the case of polynomial maps $p:\mathbb Z\rightarrow\mathbb Z$ vanishing at zero which are always good for nice recurrence, there are polynomials $p:\mathbb F_2^\omega\rightarrow \mathbb F_2^\omega$ with $p(0_{\mathbb F_2^\omega})=0_{\mathbb F_2^\omega}$ which fail to have this property. This disproves a conjecture of Bergelson and McCutcheon (c. 2000), which predicted that for any countably infinite abelian groups $H$ and $G$, every polynomial map $p:H\to G$ with $p(0_H)=0_G$ is good for nice recurrence. Moreover, we develop a dynamical mechanism which shows that the magnitude of intersections along polynomial paths is degree-sensitive (even when one considers only weakly mixing systems). Among other things, we also show that the Furstenberg-Sarkozy theorem for $\mathbb F_2^\omega$-valued polynomials of degree at most $d$ vanishing at zero is equivalent to a $d$-dimensional symmetric-difference weakening of the density polynomial Hales-Jewett conjecture. Thus, as we explain in detail in this paper, our observations not only shed new light on the phenomenon of polynomial recurrence but also constrain possible strategies for proving or disproving the density polynomial Hales-Jewett conjecture.
Let $f$ be multiplicative, with $|f(p)|\le A$ at primes and $\sum_{n\le x}|f(n)|^2\le A^2x$ for every $x\ge1$. If $|\alpha-a/q|\le q^{-2}$, $(a,q)=1$, and $3\le R\le q\le N/R$, we prove \[ \sum_{n\le N}f(n)\operatorname{e}(n\alpha) \ll_A \frac{N}{\log N} +\frac{N}{\sqrt R}\sqrt{\log\log(3R)} \] with effective implied constants. Montgomery and Vaughan proved this with second term $NR^{-1/2}(\log R)^{3/2}$, and, for $1$-bounded functions, Bachman replaced it by $NR^{-1/2}\sqrt{\log R\log\log R}$. We remove the factor $\sqrt{\log R}$ from Bachman's second term while retaining the original coefficient hypotheses of Montgomery and Vaughan. A more precise estimate records the distance from a rational number. The proof combines the Brun-Titchmarsh inequality on short intervals with maximal Fourier estimates derived from the Carleson-Hunt theorem; the local bounds permit arbitrary prime-dependent prefixes. We also prove sharpness of the square-root displacement dependence.
We study the generalized Hilbert operator \[ \mathcal{H}_b f(z)=\int_0^1 f(t)\,\frac{(1-t)^b}{(1-tz)^{b+1}}\,dt, \qquad b>0, \] acting on the Hardy spaces $H^p$ for $1\leq p\leq \infty$. We establish the precise operator norm \[ \|\mathcal{H}_b\|_{H^p\to H^p}=B\!\left(\frac1p,b+1-\frac1p\right) \] for every $1<p<\infty$ and, by continuous extension to $b=0$, recover the classical norm $\pi/\sin(\pi/p)$. We also prove that $\mathcal{H}_b$ is bounded on $H^1$ for every $b>0$, in contrast with the classical Hilbert operator, and we obtain the sharp restricted norm estimate \[ \|\mathcal{H}_b\|_{H^1_0\to H^1}=B(1,b). \] We also determine the exact norm \[ \|\mathcal{H}_b\|_{H^\infty\to \mathcal B}=\frac{1}{b+1}+2. \]
Let $F=\mathbb{F}_q((u))((t))$ and $H=\mathrm{GL}_n(F)_L\times\mathrm{GL}_n(F)_R$. An ordered tuple $\alpha=(\alpha_1,\dots, \alpha_n) $ defines a cross-$K_2$ multiplier on the doubled Borel. Twisted equivariantization then yields an exact uniformly smooth categorical principal series $2$-representation of $H$. For unitary parameters, the $\mathbb R((X))$-measure gives a positive Hom-valued formal Hermitian inner product on the spherical part. These constructions provide two-dimensional categorical analogues of selected unramified structures.
For $k\geq1$, let $M_k=[(z-w)^k]\subset H^2(\mathbb D^2)$. We first determine the banded Toeplitz matrices associated with the homogeneous components of $M_k$, together with explicit formulas for their determinants and the relevant algebraic cofactors. These formulas lead to a complete description of the spectrum of the core operator: \[ \sigma(C_{M_k}) = \{0,1\} \cup \left\{ \pm\frac{k}{n+k}:n\geq1 \right\}. \] In particular, the spectral data determine the parameter $k$. The determinant and cofactor formulas further yield a unified finite-sum representation for $\alpha_{n,j}^{(k)} =\langle w^j\phi_n,z^j\psi_n\rangle$, and hence for Yang's higher numerical invariants. We derive an adjacent relation connecting $\alpha_{n,j}^{(k)}$ and $\alpha_{n,j+1}^{(k)}$ by means of an explicit telescoping certificate, and show that the corresponding finite-section transformations are strict contractions. Combining these finite-dimensional estimates with the asymptotic behavior of $\alpha_{n,j}^{(k)}$, we prove the strict monotonicity \[ \Sigma_0(M_k)>\Sigma_1(M_k)>\Sigma_2(M_k)>\cdots . \] The cases $k\geq3$ constitute the new part of the analysis, while the previously known cases $k=1,2$ are recovered within the same framework. Consequently, Yang's monotonicity conjecture holds in strict form for the entire family $\{[(z-w)^k]:k\geq1\}$.
Yin Liu, Yufeng Lu, Chao Zu· 0 citations
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