Let $H = N^{\theta }, \theta> 2/3$ and $k \geq 1$. We obtain estimates for the following exponential sum over primes in short intervals: \begin{equation*} \sum_{N < n \leq N+H} \Lambda(n) \mathrm e(g(n)), \end{equation*}where $g$ is a polynomial of degree $k$. As a consequence of this in the special case $g(n) = \alpha n^k$, we deduce a short interval version of the Waring–Goldbach problem.
Let $\alpha \colon X \to Y$ be a general degree $r$ primitive map of nonsingular, irreducible, projective curves over an algebraically closed field of characteristic zero or larger than $r$. We prove that the Tschirnhausen bundle of $\alpha $ is semistable if $g(Y) \geq 1$ and stable if $g(Y) \geq 2$.
more » « less- NSF-PAR ID:
- 10408711
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2024
- Issue:
- 1
- ISSN:
- 1073-7928
- Format(s):
- Medium: X Size: p. 597-612
- Size(s):
- ["p. 597-612"]
- Sponsoring Org:
- National Science Foundation
More Like this
-
Abstract -
Meila, Marina ; Zhang, Tong (Ed.)In the Correlation Clustering problem, we are given a complete weighted graph $G$ with its edges labeled as “similar" and “dissimilar" by a noisy binary classifier. For a clustering $\mathcal{C}$ of graph $G$, a similar edge is in disagreement with $\mathcal{C}$, if its endpoints belong to distinct clusters; and a dissimilar edge is in disagreement with $\mathcal{C}$ if its endpoints belong to the same cluster. The disagreements vector, $\mathbf{disagree}$, is a vector indexed by the vertices of $G$ such that the $v$-th coordinate $\mathbf{disagree}_v$ equals the weight of all disagreeing edges incident on $v$. The goal is to produce a clustering that minimizes the $\ell_p$ norm of the disagreements vector for $p\geq 1$. We study the $\ell_p$ objective in Correlation Clustering under the following assumption: Every similar edge has weight in $[\alpha\mathbf{w},\mathbf{w}]$ and every dissimilar edge has weight at least $\alpha\mathbf{w}$ (where $\alpha \leq 1$ and $\mathbf{w}>0$ is a scaling parameter). We give an $O\left((\frac{1}{\alpha})^{\frac{1}{2}-\frac{1}{2p}}\cdot \log\frac{1}{\alpha}\right)$ approximation algorithm for this problem. Furthermore, we show an almost matching convex programming integrality gap.more » « less
-
Abstract We show that the energy conditions are not necessary for boundedness of Riesz transforms in dimension $n\geq 2$. In dimension $n=1$, we construct an elliptic singular integral operator $H_{\flat } $ for which the energy conditions are not necessary for boundedness of $H_{\flat }$. The convolution kernel $K_{\flat }\left ( x\right ) $ of the operator $H_{\flat }$ is a smooth flattened version of the Hilbert transform kernel $K\left ( x\right ) =\frac{1}{x}$ that satisfies ellipticity $ \vert K_{\flat }\left ( x\right ) \vert \gtrsim \frac{1}{\left \vert x\right \vert }$, but not gradient ellipticity $ \vert K_{\flat }^{\prime }\left ( x\right ) \vert \gtrsim \frac{1}{ \vert x \vert ^{2}}$. Indeed the kernel has flat spots where $K_{\flat }^{\prime }\left ( x\right ) =0$ on a family of intervals, but $K_{\flat }^{\prime }\left ( x\right ) $ is otherwise negative on $\mathbb{R}\setminus \left \{ 0\right \} $. On the other hand, if a one-dimensional kernel $K\left ( x,y\right ) $ is both elliptic and gradient elliptic, then the energy conditions are necessary, and so by our theorem in [30], the $T1$ theorem holds for such kernels on the line. This paper includes results from arXiv:16079.06071v3 and arXiv:1801.03706v2.
-
Abstract This paper will study almost everywhere behaviors of functions on partition spaces of cardinals possessing suitable partition properties. Almost everywhere continuity and monotonicity properties for functions on partition spaces will be established. These results will be applied to distinguish the cardinality of certain subsets of the power set of partition cardinals.
The following summarizes the main results proved under suitable partition hypotheses.
If
is a cardinal,$\kappa $ ,$\epsilon < \kappa $ ,${\mathrm {cof}}(\epsilon ) = \omega $ and$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$ , then$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$ satisfies the almost everywhere short length continuity property: There is a club$\Phi $ and a$C \subseteq \kappa $ so that for all$\delta < \epsilon $ , if$f,g \in [C]^\epsilon _*$ and$f \upharpoonright \delta = g \upharpoonright \delta $ , then$\sup (f) = \sup (g)$ .$\Phi (f) = \Phi (g)$ If
is a cardinal,$\kappa $ is countable,$\epsilon $ holds and$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$ , then$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$ satisfies the strong almost everywhere short length continuity property: There is a club$\Phi $ and finitely many ordinals$C \subseteq \kappa $ so that for all$\delta _0, ..., \delta _k \leq \epsilon $ , if for all$f,g \in [C]^\epsilon _*$ ,$0 \leq i \leq k$ , then$\sup (f \upharpoonright \delta _i) = \sup (g \upharpoonright \delta _i)$ .$\Phi (f) = \Phi (g)$ If
satisfies$\kappa $ ,$\kappa \rightarrow _* (\kappa )^\kappa _2$ and$\epsilon \leq \kappa $ , then$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$ satisfies the almost everywhere monotonicity property: There is a club$\Phi $ so that for all$C \subseteq \kappa $ , if for all$f,g \in [C]^\epsilon _*$ ,$\alpha < \epsilon $ , then$f(\alpha ) \leq g(\alpha )$ .$\Phi (f) \leq \Phi (g)$ Suppose dependent choice (
),$\mathsf {DC}$ and the almost everywhere short length club uniformization principle for${\omega _1} \rightarrow _* ({\omega _1})^{\omega _1}_2$ hold. Then every function${\omega _1}$ satisfies a finite continuity property with respect to closure points: Let$\Phi : [{\omega _1}]^{\omega _1}_* \rightarrow {\omega _1}$ be the club of$\mathfrak {C}_f$ so that$\alpha < {\omega _1}$ . There is a club$\sup (f \upharpoonright \alpha ) = \alpha $ and finitely many functions$C \subseteq {\omega _1}$ so that for all$\Upsilon _0, ..., \Upsilon _{n - 1} : [C]^{\omega _1}_* \rightarrow {\omega _1}$ , for all$f \in [C]^{\omega _1}_*$ , if$g \in [C]^{\omega _1}_*$ and for all$\mathfrak {C}_g = \mathfrak {C}_f$ ,$i < n$ , then$\sup (g \upharpoonright \Upsilon _i(f)) = \sup (f \upharpoonright \Upsilon _i(f))$ .$\Phi (g) = \Phi (f)$ Suppose
satisfies$\kappa $ for all$\kappa \rightarrow _* (\kappa )^\epsilon _2$ . For all$\epsilon < \kappa $ ,$\chi < \kappa $ does not inject into$[\kappa ]^{<\kappa }$ , the class of${}^\chi \mathrm {ON}$ -length sequences of ordinals, and therefore,$\chi $ . As a consequence, under the axiom of determinacy$|[\kappa ]^\chi | < |[\kappa ]^{<\kappa }|$ , these two cardinality results hold when$(\mathsf {AD})$ is one of the following weak or strong partition cardinals of determinacy:$\kappa $ ,${\omega _1}$ ,$\omega _2$ (for all$\boldsymbol {\delta }_n^1$ ) and$1 \leq n < \omega $ (assuming in addition$\boldsymbol {\delta }^2_1$ ).$\mathsf {DC}_{\mathbb {R}}$ -
The classic graphical Cheeger inequalities state that if $M$ is an $n\times n$ \emph{symmetric} doubly stochastic matrix, then \[ \frac{1-\lambda_{2}(M)}{2}\leq\phi(M)\leq\sqrt{2\cdot(1-\lambda_{2}(M))} \] where $\phi(M)=\min_{S\subseteq[n],|S|\leq n/2}\left(\frac{1}{|S|}\sum_{i\in S,j\not\in S}M_{i,j}\right)$ is the edge expansion of $M$, and $\lambda_{2}(M)$ is the second largest eigenvalue of $M$. We study the relationship between $\phi(A)$ and the spectral gap $1-\re\lambda_{2}(A)$ for \emph{any} doubly stochastic matrix $A$ (not necessarily symmetric), where $\lambda_{2}(A)$ is a nontrivial eigenvalue of $A$ with maximum real part. Fiedler showed that the upper bound on $\phi(A)$ is unaffected, i.e., $\phi(A)\leq\sqrt{2\cdot(1-\re\lambda_{2}(A))}$. With regards to the lower bound on $\phi(A)$, there are known constructions with \[ \phi(A)\in\Theta\left(\frac{1-\re\lambda_{2}(A)}{\log n}\right), \] indicating that at least a mild dependence on $n$ is necessary to lower bound $\phi(A)$. In our first result, we provide an \emph{exponentially} better construction of $n\times n$ doubly stochastic matrices $A_{n}$, for which \[ \phi(A_{n})\leq\frac{1-\re\lambda_{2}(A_{n})}{\sqrt{n}}. \] In fact, \emph{all} nontrivial eigenvalues of our matrices are $0$, even though the matrices are highly \emph{nonexpanding}. We further show that this bound is in the correct range (up to the exponent of $n$), by showing that for any doubly stochastic matrix $A$, \[ \phi(A)\geq\frac{1-\re\lambda_{2}(A)}{35\cdot n}. \] As a consequence, unlike the symmetric case, there is a (necessary) loss of a factor of $n^{\alpha}$ for $\frac{1}{2}\leq\alpha\leq1$ in lower bounding $\phi$ by the spectral gap in the nonsymmetric setting. Our second result extends these bounds to general matrices $R$ with nonnegative entries, to obtain a two-sided \emph{gapped} refinement of the Perron-Frobenius theorem. Recall from the Perron-Frobenius theorem that for such $R$, there is a nonnegative eigenvalue $r$ such that all eigenvalues of $R$ lie within the closed disk of radius $r$ about $0$. Further, if $R$ is irreducible, which means $\phi(R)>0$ (for suitably defined $\phi$), then $r$ is positive and all other eigenvalues lie within the \textit{open} disk, so (with eigenvalues sorted by real part), $\re\lambda_{2}(R)
more » « less