We examine correlations of the Möbius function over $\mathbb{F}_{q}[t]$ with linear or quadratic phases, that is, averages of the form 1 $$\begin{eqnarray}\frac{1}{q^{n}}\mathop{\sum }_{\deg f0$ if $Q$ is linear and $O(q^{-n^{c}})$ for some absolute constant $c>0$ if $Q$ is quadratic. The latter bound may be reduced to $O(q^{-c^{\prime }n})$ for some $c^{\prime }>0$ when $Q(f)$ is a linear form in the coefficients of $f^{2}$ , that is, a Hankel quadratic form, whereas, for general quadratic forms, it relies on a bilinear version of the additive-combinatorial Bogolyubov theorem.
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CUTTING A PART FROM MANY MEASURES
Holmsen, Kynčl and Valculescu recently conjectured that if a finite set $X$ with $\ell n$ points in $\mathbb{R}^{d}$ that is colored by $m$ different colors can be partitioned into $n$ subsets of $\ell$ points each, such that each subset contains points of at least $d$ different colors, then there exists such a partition of $X$ with the additional property that the convex hulls of the $n$ subsets are pairwise disjoint. We prove a continuous analogue of this conjecture, generalized so that each subset contains points of at least $c$ different colors, where we also allow $c$ to be greater than $d$ . Furthermore, we give lower bounds on the fraction of the points each of the subsets contains from $c$ different colors. For example, when $n\geqslant 2$ , $d\geqslant 2$ , $c\geqslant d$ with $m\geqslant n(c-d)+d$ are integers, and $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{m}$ are $m$ positive finite absolutely continuous measures on $\mathbb{R}^{d}$ , we prove that there exists a partition of $\mathbb{R}^{d}$ into $n$ convex pieces which equiparts the measures $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{d-1}$ , and in addition every piece of the partition has positive measure with respect to at least $c$ of the measures $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{m}$ .
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- Award ID(s):
- 1851420
- NSF-PAR ID:
- 10157302
- Date Published:
- Journal Name:
- Forum of Mathematics, Sigma
- Volume:
- 7
- ISSN:
- 2050-5094
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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For each $t\in \mathbb{R}$ , we define the entire function $$\begin{eqnarray}H_{t}(z):=\int _{0}^{\infty }e^{tu^{2}}\unicode[STIX]{x1D6F7}(u)\cos (zu)\,du,\end{eqnarray}$$ where $\unicode[STIX]{x1D6F7}$ is the super-exponentially decaying function $$\begin{eqnarray}\unicode[STIX]{x1D6F7}(u):=\mathop{\sum }_{n=1}^{\infty }(2\unicode[STIX]{x1D70B}^{2}n^{4}e^{9u}-3\unicode[STIX]{x1D70B}n^{2}e^{5u})\exp (-\unicode[STIX]{x1D70B}n^{2}e^{4u}).\end{eqnarray}$$ Newman showed that there exists a finite constant $\unicode[STIX]{x1D6EC}$ (the de Bruijn–Newman constant ) such that the zeros of $H_{t}$ are all real precisely when $t\geqslant \unicode[STIX]{x1D6EC}$ . The Riemann hypothesis is equivalent to the assertion $\unicode[STIX]{x1D6EC}\leqslant 0$ , and Newman conjectured the complementary bound $\unicode[STIX]{x1D6EC}\geqslant 0$ . In this paper, we establish Newman’s conjecture. The argument proceeds by assuming for contradiction that $\unicode[STIX]{x1D6EC}<0$ and then analyzing the dynamics of zeros of $H_{t}$ (building on the work of Csordas, Smith and Varga) to obtain increasingly strong control on the zeros of $H_{t}$ in the range $\unicode[STIX]{x1D6EC}Abstract Let M be a complete Riemannian manifold and suppose {p\in M} . For each unit vector {v\in T_{p}M} , the Jacobi operator , {\mathcal{J}_{v}:v^{\perp}\rightarrow v^{\perp}} is the symmetric endomorphism, {\mathcal{J}_{v}(w)=R(w,v)v} . Then p is an isotropic point if there exists a constant {\kappa_{p}\in{\mathbb{R}}} such that {\mathcal{J}_{v}=\kappa_{p}\operatorname{Id}_{v^{\perp}}} for each unit vector {v\in T_{p}M} . If all points are isotropic, then M is said to be isotropic; it is a classical result of Schur that isotropic manifolds of dimension at least 3 have constant sectional curvatures. In this paper we consider almost isotropic manifolds , i.e. manifolds having the property that for each {p\in M} , there exists a constant {\kappa_{p}\in\mathbb{R}} such that the Jacobi operators {\mathcal{J}_{v}} satisfy {\operatorname{rank}({\mathcal{J}_{v}-\kappa_{p}\operatorname{Id}_{v^{\perp}}}% )\leq 1} for each unit vector {v\in T_{p}M} . Our main theorem classifies the almost isotropic simply connected Kähler manifolds, proving that those of dimension {d=2n\geqslant 4} are either isometric to complex projective space or complex hyperbolic space or are totally geodesically foliated by leaves isometric to {{\mathbb{C}}^{n-1}} .more » « lessnull (Ed.)Let $f:X\rightarrow X$ be a continuous dynamical system on a compact metric space $X$ and let $\unicode[STIX]{x1D6F7}:X\rightarrow \mathbb{R}^{m}$ be an $m$ -dimensional continuous potential. The (generalized) rotation set $\text{Rot}(\unicode[STIX]{x1D6F7})$ is defined as the set of all $\unicode[STIX]{x1D707}$ -integrals of $\unicode[STIX]{x1D6F7}$ , where $\unicode[STIX]{x1D707}$ runs over all invariant probability measures. Analogous to the classical topological entropy, one can associate the localized entropy $\unicode[STIX]{x210B}(w)$ to each $w\in \text{Rot}(\unicode[STIX]{x1D6F7})$ . In this paper, we study the computability of rotation sets and localized entropy functions by deriving conditions that imply their computability. Then we apply our results to study the case where $f$ is a subshift of finite type. We prove that $\text{Rot}(\unicode[STIX]{x1D6F7})$ is computable and that $\unicode[STIX]{x210B}(w)$ is computable in the interior of the rotation set. Finally, we construct an explicit example that shows that, in general, $\unicode[STIX]{x210B}$ is not continuous on the boundary of the rotation set when considered as a function of $\unicode[STIX]{x1D6F7}$ and $w$ . In particular, $\unicode[STIX]{x210B}$ is, in general, not computable at the boundary of $\text{Rot}(\unicode[STIX]{x1D6F7})$ .more » « lessLet $\unicode[STIX]{x1D6FC}\in \mathbb{R}\backslash \mathbb{Q}$ and $\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})=\limsup _{n\rightarrow \infty }(\ln q_{n+1})/q_{n}<\infty$ , where $p_{n}/q_{n}$ is the continued fraction approximation to $\unicode[STIX]{x1D6FC}$ . Let $(H_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}}u)(n)=u(n+1)+u(n-1)+2\unicode[STIX]{x1D706}\cos 2\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D703}+n\unicode[STIX]{x1D6FC})u(n)$ be the almost Mathieu operator on $\ell ^{2}(\mathbb{Z})$ , where $\unicode[STIX]{x1D706},\unicode[STIX]{x1D703}\in \mathbb{R}$ . Avila and Jitomirskaya [The ten Martini problem. Ann. of Math. (2), 170 (1) (2009), 303–342] conjectured that, for $2\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6FC}\mathbb{Z}+\mathbb{Z}$ , $H_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}}$ satisfies Anderson localization if $|\unicode[STIX]{x1D706}|>e^{2\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})}$ . In this paper, we develop a method to treat simultaneous frequency and phase resonances and obtain that, for $2\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6FC}\mathbb{Z}+\mathbb{Z}$ , $H_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}}$ satisfies Anderson localization if $|\unicode[STIX]{x1D706}|>e^{3\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})}$ .more » « less
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/fms.2019.33
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