Asymptotic diophantine approximation: the multiplicative case

Let \(\alpha \) and \(\beta \) be irrational real numbers and \(0<\varepsilon <1/30\). We prove a precise estimate for the number of positive integers \(q\le Q\) that satisfy \(\Vert q\alpha \Vert \cdot \Vert q\beta \Vert <\varepsilon \). If we choose \(\varepsilon \) as a function of Q, we get asymptotics as Q gets large, provided \(\varepsilon Q\) grows quickly enough in terms of the (multiplicative) Diophantine type of \((\alpha ,\beta )\), e.g., if \((\alpha ,\beta )\) is a counterexample to Littlewood’s conjecture, then we only need that \(\varepsilon Q\) tends to infinity. Our result yields a new upper bound on sums of reciprocals of products of fractional parts and sheds some light on a recent question of Lê and Vaaler.     

Closure properties of parametric subcompleteness

For an ordinal \(\varepsilon \), I introduce a variant of the notion of subcompleteness of a forcing poset, which I call \(\varepsilon \)-subcompleteness, and show that this class of forcings enjoys some closure properties that the original class of subcomplete forcings does not seem to have: factors of \(\varepsilon \)-subcomplete forcings are \(\varepsilon \)-subcomplete, and if \(\mathbb {P}\) and \(\mathbb {Q}\) are forcing-equivalent notions, then \(\mathbb {P}\) is \(\varepsilon \)-subcomplete iff \(\mathbb {Q}\) is. I formulate a Two Step Theorem for \(\varepsilon \)-subcompleteness and prove an RCS iteration theorem for \(\varepsilon \)-subcompleteness which is slightly less restrictive than the original one, in that its formulation is more careful about the amount of collapsing necessary. Finally, I show that an adequate degree of \(\varepsilon \)-subcompleteness follows from the \(\kappa \)-distributivity of a forcing, for \(\kappa >\omega _1\).     

Edge Flips in Surface Meshes

Little theoretical work has been done on edge flips in surface meshes despite their popular usage in graphics and solid modeling to improve mesh equality. We propose the class of \((\varepsilon ,\alpha )\)-meshes of a surface that satisfy several properties: the vertex set is an \(\varepsilon \)-sample of the surface, the triangle angles are no smaller than a constant \(\alpha \), some triangle has a good normal, and the mesh is homeomorphic to the surface. We believe that many surface meshes encountered in practice are \((\varepsilon ,\alpha )\)-meshes or close to being one. We prove that flipping the appropriate edges can smooth a dense \((\varepsilon ,\alpha )\)-mesh by making the triangle normals better approximations of the surface normals and the dihedral angles closer to \(\pi \). Moreover, the edge flips can be performed in time linear in the number of vertices. This helps to explain the effectiveness of edge flips as observed in practice and in our experiments. A corollary of our techniques is that, in \(\mathbb {R}^2\), every triangulation with a constant lower bound on the angles can be flipped in linear time to the Delaunay triangulation.     

General Natural $$(\alpha ,\varepsilon )$$-Structures

We study in a unified way the \((\alpha ,\varepsilon )\)-structures of general natural lift type on the tangent bundle of a Riemannian manifold. We characterize the general natural \(\alpha \)-structures on the total space of the tangent bundle of a Riemannian manifold, and provide their integrability conditions (the base manifold is a space form and some involved coefficients are rational functions of the other ones). Then, we characterize the two classes (with respect to the sign of \(\alpha \varepsilon \)) of \((\alpha ,\varepsilon )\)-structures of general natural type on TM. The class \(\alpha \varepsilon =-1\) is characterized by some proportionality relations between the coefficients of the metric and those of the \(\alpha \)-structure, and in this case, the structure is almost Kählerian if and only if the first proportionality factor is the derivative of the second one. Moreover, the total space of the tangent bundle is a Kähler manifold if and only if it depends on three coefficients only (two coefficients of the integrable \(\alpha \)-structure and a proportionality factor).     

Improved bounds on equilibria solutions in the network design game

In the network design game with n players, every player chooses a path in an edge-weighted graph to connect her pair of terminals, sharing costs of the edges on her path with all other players fairly. It has been shown that the price of stability of any network design game is at most \(H_n\), the n-th harmonic number. This bound is tight for directed graphs.For undirected graphs, it has only recently been shown that the price of stability is at most \(H_n \left( 1-\frac{1}{\Theta (n^4)} \right) \), while the worst-case known example has price of stability around 2.25. We improve the upper bound considerably by showing that the price of stability is at most \(H_{n/2} + \varepsilon \) for any \(\varepsilon \) starting from some suitable \(n \ge n(\varepsilon )\).We also study quality measures of different solution concepts for the multicast network design game on a ring topology. We recall from the literature a lower bound of \(\frac{4}{3}\) and prove a matching upper bound for the price of stability. Therefore, we answer an open question posed by Fanelli et al. (Theor Comput Sci 562:90–100, 2015). We prove an upper bound of 2 for the ratio of the costs of a potential optimizer and of an optimum, provide a construction of a lower bound, and give a computer-assisted argument that it reaches 2 for any precision. We then turn our attention to players arriving one by one and playing myopically their best response. We provide matching lower and upper bounds of 2 for the myopic sequential price of anarchy (achieved for a worst-case order of the arrival of the players). We then initiate the study of myopic sequential price of stability and for the multicast game on the ring we construct a lower bound of \(\frac{4}{3}\), and provide an upper bound of \(\frac{26}{19}\). To the end, we conjecture and argue that the right answer is \(\frac{4}{3}\).     

Existence of self-Cheeger sets on Riemannian manifolds

Let \(({\mathcal M},g)\) be a smooth compact Riemannian manifold of dimension \(N\ge 2\). We prove the existence of a family \((\Omega _\varepsilon )_{\varepsilon \in (0,\varepsilon _0)}\) of self-Cheeger sets in \(({\mathcal M},g)\). The domains \(\Omega _\varepsilon \subset {\mathcal M}\) are perturbations of geodesic balls of radius \(\varepsilon \) centered at \(p \in {\mathcal M}\), and in particular, if \(p_0\) is a non-degenerate critical point of the scalar curvature of g, then the family \((\partial \Omega _\varepsilon )_{\varepsilon \in (0,\varepsilon _0)}\) constitutes a smooth foliation of a neighborhood of \(p_0\).     

Fractal Percolation,Porosity, and Dimension

We study the porosity properties of fractal percolation sets \(E\subset \mathbb {R}^d\). Among other things, for all \(0<\varepsilon <\tfrac{1}{2}\), we obtain dimension bounds for the set of exceptional points where the upper porosity of E is less than \(\tfrac{1}{2}-\varepsilon \), or the lower porosity is larger than \(\varepsilon \). Our method works also for inhomogeneous fractal percolation and more general random sets whose offspring distribution gives rise to a Galton–Watson process.     

The Sobolev Stability Threshold for 2D Shear Flows Near Couette

We consider the 2D Navier–Stokes equation on \(\mathbb T \times \mathbb R\), with initial datum that is \(\varepsilon \)-close in \(H^N\) to a shear flow (U(y), 0), where \(\Vert U(y) - y\Vert _{H^{N+4}} \ll 1\) and \(N>1\). We prove that if \(\varepsilon \ll \nu ^{1/2}\), where \(\nu \) denotes the inverse Reynolds number, then the solution of the Navier–Stokes equation remains \(\varepsilon \)-close in \(H^1\) to \((e^{t \nu \partial _{yy}}U(y),0)\) for all \(t>0\). Moreover, the solution converges to a decaying shear flow for times \(t \gg \nu ^{-1/3}\) by a mixing-enhanced dissipation effect, and experiences a transient growth of gradients. In particular, this shows that the stability threshold in finite regularity scales no worse than \(\nu ^{1/2}\) for 2D shear flows close to the Couette flow.     

The Allen–Cahn equation on closed manifolds

We study global variational properties of the space of solutions to \(-\varepsilon ^2\Delta u + W'(u)=0\) on any closed Riemannian manifold M. Our techniques are inspired by recent advances in the variational theory of minimal hypersurfaces and extend a well-known analogy with the theory of phase transitions. First, we show that solutions at the lowest positive energy level are either stable or obtained by min–max and have index 1. We show that if \(\varepsilon \) is not small enough, in terms of the Cheeger constant of M, then there are no interesting solutions. However, we show that the number of min–max solutions to the equation above goes to infinity as \(\varepsilon \rightarrow 0\) and their energies have sublinear growth. This result is sharp in the sense that for generic metrics the number of solutions is finite, for fixed \(\varepsilon \), as shown recently by G. Smith. We also show that the energy of the min–max solutions accumulate, as \(\varepsilon \rightarrow 0\), around limit-interfaces which are smooth embedded minimal hypersurfaces whose area with multiplicity grows sublinearly. For generic metrics with \(\mathrm{Ric}_M>0\), the limit-interface of the solutions at the lowest positive energy level is an embedded minimal hypersurface of least area in the sense of Mazet–Rosenberg. Finally, we prove that the min–max energy values are bounded from below by the widths of the area functional as defined by Marques–Neves.     

Approximation,solution operators and quantale-valued metrics

A generalized solution operator is a mapping abstractly describing a computational problem and its approximate solutions. It assigns a set of \(\varepsilon \)-approximations of a solution to the problem instance f and accuracy of approximation \(\varepsilon \). In this paper we study generalized solution operators for which the accuracy of approximation is described by elements of a complete lattice equipped with a compatible monoid structure, namely, a quantale. We provide examples of computational problems for which the accuracy of approximation of a solution is measured by such objects. We show that the sets of \(\varepsilon \)-approximations are, roughly, closed balls with radii \(\varepsilon \) with respect to a certain family of quantale-valued generalized metrics induced by a generalized solution operator.     

rthogonality Property $$\mathcal {}$$ and Compact Perturbations

A bounded linear operator T acting on a Hilbert space is said to have orthogonality property \(\mathcal {O}\) if the subspaces \(\ker (T-\alpha )\) and \(\ker (T-\beta )\) are orthogonal for all \(\alpha , \beta \in \sigma _p(T)\) with \(\alpha \ne \beta \). In this paper, the authors investigate the compact perturbations of operators with orthogonality property \(\mathcal {O}\). We give a sufficient and necessary condition to determine when an operator T has the following property: for each \(\varepsilon >0\), there exists \(K\in \mathcal {K(H)}\) with \(\Vert K\Vert <\varepsilon \) such that \(T+K\) has orthogonality property \(\mathcal {O}\). Also, we study the stability of orthogonality property \(\mathcal {O}\) under small compact perturbations and analytic functional calculus.     

Almost band preservers

We study the stability of band preserving operators on Banach lattices. To this end the notion of \(\varepsilon \)-band preserving mapping is introduced. It is shown that, under quite general assumptions, a \(\varepsilon \)-band preserving operator is in fact a small perturbation of a band preserving one. However, a counterexample can be produced in some circumstances. Some results on automatic continuity of \(\varepsilon \)-band preserving maps are also obtained.     

Construction of sign-changing solutions for a harmonic equation with subcritical exponent

In this paper, we study the harmonic equation involving subcritical exponent \((P_{\varepsilon })\): \( \Delta u = 0 \), in \(\mathbb {B}^n\) and \(\displaystyle \frac{\partial u}{\partial \nu } + \displaystyle \frac{n-2}{2}u = \displaystyle \frac{n-2}{2} K u^{\frac{n}{n-2}-\varepsilon }\) on \( \mathbb {S}^{n-1}\) where \(\mathbb {B}^n \) is the unit ball in \(\mathbb {R}^n\), \(n\ge 5\) with Euclidean metric \(g_0\), \(\partial \mathbb {B}^n = \mathbb {S}^{n-1}\) is its boundary, K is a function on \(\mathbb {S}^{n-1}\) and \(\varepsilon \) is a small positive parameter. We construct solutions of the subcritical equation \((P_{\varepsilon })\) which blow up at two different critical points of K. Furthermore, we construct solutions of \((P_{\varepsilon })\) which have two bubbles and blow up at the same critical point of K.     

A note on <Emphasis Type="Italic">n</Emphasis>! modulo <Emphasis Type="Italic">p</Emphasis>

Let p be a prime, \(\varepsilon >0\) and \(0<L+1<L+N < p\). We prove that if \(p^{1/2+\varepsilon }< N <p^{1-\varepsilon }\), then

$$\begin{aligned} \#\{n!\,\,({\mathrm{mod}} \,p);\,\, L+1\le n\le L+N\} > c (N\log N)^{1/2},\,\, c=c(\varepsilon )>0. \end{aligned}$$

We use this bound to show that any \(\lambda \not \equiv 0\ ({\mathrm{mod}}\, p)\) can be represented in the form \(\lambda \equiv n_1!\cdots n_7!\ ({\mathrm{mod}}\, p)\), where \(n_i=o(p^{11/12})\). This refines the previously known range for \(n_i\).

     

Associate space with respect to a locally $$\sigma $$-finite measure on a $$\delta $$δ-ring and applications to spaces of integrable functions defined by a vector measure

We show that for a locally \(\sigma \)-finite measure \(\mu \) defined on a \(\delta \)-ring, the associate space theory can be developed as in the \(\sigma \)-finite case, and corresponding properties are obtained. Given a saturated \(\sigma \)-order continuous \(\mu \)-Banach function space E, we prove that its dual space can be identified with the associate space \(E ^\times \) if, and only if, \(E^\times \) has the Fatou property. Applying the theory to the spaces \(L^p (\nu )\) and \(L_w^p (\nu )\), where \(\nu \) is a vector measure defined on a \(\delta \)-ring \(\mathcal {R}\) and \(1 \le p < \infty \), we establish results corresponding to those of the case when the vector measure is defined on a \(\sigma \)-algebra.     

Eichler’s commutation relation and some other invariant subspaces of Hecke operators

Let k be an odd positive integer, L a lattice on a regular positive definite k-dimensional quadratic space over \(\mathbb {Q}\), \(N_L\) the level of L, and \(\mathscr {M}(L)\)  be the linear space of \(\theta \)-series attached to the distinct classes in the genus of L. We prove that, for an odd prime \(p|N_L\), if \(L_p=L_{p,1}\,\bot \, L_{p,2}\), where \(L_{p,1}\) is unimodular, \(L_{p,2}\) is (p)-modular, and \(\mathbb {Q}_pL_{p,2}\) is anisotropic, then \(\mathscr {M}(L;p):=\) \(\mathscr {M}(L)\) \(+T_{p^2}.\) \(\mathscr {M}(L)\)  is stable under the Hecke operator \(T_{p^2}\). If \(L_2\) is isometric to \(\left( \begin{array}{ll}0&{}\frac{1}{2}\\ \frac{1}{2}&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle \varepsilon \rangle \) or \(\left( \begin{array}{ll}0&{}\frac{1}{2}\\ \frac{1}{2}&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle 2\varepsilon \rangle \) or \(\left( \begin{array}{ll}0&{}1\\ 1&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle \varepsilon \rangle \) with \(\varepsilon \in \mathbb {Z}_2^{\times }\) and \(\kappa :=\frac{k-1}{2}\), then \(\mathscr {M}(L;2):=T_{2^2}.\mathscr {M}(L)+T_{2^2}^2.\,\mathscr {M}(L)\) is stable under the Hecke operator \(T_{2^2}\). Furthermore, we determine some invariant subspaces of the cusp forms for the Hecke operators.     

Octahedral developing of knot complement I: Pseudo-hyperbolic structure

The width w of a curve \(\gamma \) in Euclidean space \({\mathbf {R}}^n\) is the infimum of the distances between all pairs of parallel hyperplanes which bound \(\gamma \), while its inradius r is the supremum of the radii of all spheres which are contained in the convex hull of \(\gamma \) and are disjoint from \(\gamma \). We use a mixture of topological and integral geometric techniques, including an application of Borsuk Ulam theorem due to Wienholtz and Crofton’s formulas, to obtain lower bounds on the length of \(\gamma \) subject to constraints on r and w. The special case of closed curves is also considered in each category. Our estimates confirm some conjectures of Zalgaller up to 99% of their stated value, while we also disprove one of them.     

Toric Aspects of the First Eigenvalue

In this paper we study the smallest non-zero eigenvalue \(\lambda _1\) of the Laplacian on toric Kähler manifolds. We find an explicit upper bound for \(\lambda _1\) in terms of moment polytope data. We show that this bound can only be attained for \(\mathbb C\mathbb P^n\) endowed with the Fubini–Study metric and therefore \(\mathbb C\mathbb P^n\) endowed with the Fubini–Study metric is spectrally determined among all toric Kähler metrics. We also study the equivariant counterpart of \(\lambda _1\) which we denote by \(\lambda _1^T\). It is the smallest non-zero eigenvalue of the Laplacian restricted to torus-invariant functions. We prove that \(\lambda _1^T\) is not bounded among toric Kähler metrics thus generalizing a result of Abreu–Freitas on \(S^2\). In particular, \(\lambda _1^T\) and \(\lambda _1\) do not coincide in general.     

The Sharp Weighted Bound for Multilinear Maximal Functions and Calderón–Zygmund Operators

We investigate the weighted bounds for multilinear maximal functions and Calderón–Zygmund operators from \(L^{p_1}(w_1)\times \cdots \times L^{p_m}(w_m)\) to \(L^{p}(v_{\vec {w}})\), where \(1<p_1,\cdots ,p_m<\infty \) with \(1/{p_1}+\cdots +1/{p_m}=1/p\) and \(\vec {w}\) is a multiple \(A_{\vec {P}}\) weight. We prove the sharp bound for the multilinear maximal function for all such \(p_1,\ldots , p_m\) and prove the sharp bound for \(m\)-linear Calderón–Zymund operators when \(p\ge 1\).     

Polynomial Carleson Operators Along Monomial Curves in the Plane

We prove \(L^p\) bounds for partial polynomial Carleson operators along monomial curves \((t,t^m)\) in the plane \(\mathbb {R}^2\) with a phase polynomial consisting of a single monomial. These operators are “partial” in the sense that we consider linearizing stopping-time functions that depend on only one of the two ambient variables. A motivation for studying these partial operators is the curious feature that, despite their apparent limitations, for certain combinations of curve and phase, \(L^2\) bounds for partial operators along curves imply the full strength of the \(L^2\) bound for a one-dimensional Carleson operator, and for a quadratic Carleson operator.