Alex Iosevich, Nets Katz, Terry Tao
Let $Ω$ be a compact convex domain in the plane. We prove that $L^2(Ω)$ has an orthogonal basis of exponentials if and only if $Ω$ tiles the plane by translation.
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Let $Ω$ be a convex polytope in $\mathbb{R}^d$. We say that $Ω$ is spectral if the space $L^2(Ω)$ admits an orthogonal basis consisting of exponential functions. There is a conjecture, which goes back to Fuglede (1974), that $Ω$ is spectral if and only if it can tile the space by translations. It is known that if $Ω$ tiles then it is spectral, but the converse was proved only in dimension $d=2$, by Iosevich, Katz and Tao. By a result due to Kolountzakis, if a convex polytope $Ω\subset \mathbb{R}^d$ is spectral, then it must be centrally symmetric. We prove that also all the facets of $Ω$ are c…
A set $Ω\subset \mathbb{R}^d$ is said to be spectral if the space $L^2(Ω)$ has an orthogonal basis of exponential functions. It is well-known that in many respects, spectral sets "behave like" sets which can tile the space by translations. This suggests a conjecture that a product set $Ω= A \times B$ is spectral if and only if the factors $A$ and $B$ are both spectral sets. We recently proved this in the case when $A$ is an interval in dimension one. The main result of the present paper is that the conjecture is true also when $A$ is a convex polygon in two dimensions. We discuss this result i…
A set $Ω\subset \mathbb{R}^d$ is said to be spectral if the space $L^2(Ω)$ has an orthogonal basis of exponential functions. A conjecture due to Fuglede (1974) stated that $Ω$ is a spectral set if and only if it can tile the space by translations. While this conjecture was disproved for general sets, it has long been known that for a convex body $Ω\subset \mathbb{R}^d$ the "tiling implies spectral" part of the conjecture is in fact true. To the contrary, the "spectral implies tiling" direction of the conjecture for convex bodies was proved only in $\mathbb{R}^2$, and also in $\mathbb{R}^3$ und…
A conjecture of Fuglede states that a bounded measurable set $Ω$ in space, of measure 1, can tile space by translations if and only if the Hilbert space $L^2(Ω)$ has an orthonormal basis consisting of exponentials. If $Ω$ has the latter property it is called spectral. We generalize a result of Fuglede, that a triangle in the plane is not spectral, proving that every non-symmetric convex domain is not spectral.
We prove that no smooth symmetric convex body $Ω$ with at least one point of non-vanishing Gaussian curvature can admit an orthogonal basis of exponentials. (The non-symmetric case was proven by Kolountzakis). This is further evidence of Fuglede's conjecture, which states that such a basis is possible if and only if $Ω$ can tile $R^d$ by translations.
A conjecture of Fuglede states that a bounded measurable set D, of measure 1, can tile space by translations if and only if the Hilbert space L^2(D) has an orthonormal basis consisting of exponentials exp(i 2 pi lambda x). If D has the latter property it is called spectral. Let D be a polytope with the following property: there is a direction u such that, of all the polytope faces perpendicular to u, the total area of the faces pointing in the positive u direction is more than the total area of the faces pointing in the negative u direction. It is almost obvious that such a polytope D cannot t…
Spectral sets and weak tiling
本文研究了谱集几何必要条件“弱平铺”的性质，并证明其在凸体、非凸多胞体、乘积域及正测度Cantor集上的应用，深化了对Fuglede猜想的理解。
For arbitrary planar convex domain, we compute the behavior of Christoffel function up to a constant factor using comparison with other simple reference domains. The lower bound is obtained by constructing an appropriate ellipse contained in the domain, while for the upper bound an appropriate parallelepiped containing the domain is constructed. As an application we obtain a new proof of existence of optimal polynomial meshes in planar convex domains.
We study convexity properties of energy functions in plane nonlinear elasticity of incompressible materials and show that rank-one convexity of an objective and isotropic elastic energy $W$ on the special linear group $\mathrm{SL}(2)$ implies the polyconvexity of $W$.
We prove a general lower bound on Christoffel function on planar convex domains in terms of a modification of the parallel section function of the domain. For a certain class of planar convex domains, in combination with a recent general upper bound, this allows to compute the pointwise behavior of Christoffel function. We illustrate this approach for the domains $\{(x,y):|x|^α+|y|^α\le1\}$, $1<α<2$, and compute up to a constant factor the required modification of the parallel section function, and, consequently, Christoffel function at an arbitrary interior point of the domain.
Uniformly perfect measures on strictly convex planar graphs are $L^{2}$-flattening
本文针对严格凸平面$C^2$图上的一致完美测度展开研究，证明该类测度均满足$L^2$平坦性质，给出了对应傅里叶变换的$L^p$范数衰减估计。
The aim of this paper is to prove that if a planar set $A$ has a difference set $Δ(A)$ satisfying $Δ(A)\subset \Z^++s$ for suitable $s$ than $A$ has at most 3 elements. This result is motivated by the conjecture that the disk has not more than 3 orthogonal exponentials. Further, we prove that if $A$ is a set of exponentials mutually orthogonal with respect to any symmetric convex set $K$ in the plane with a smooth boundary and everywhere non-vanishing curvature, then $ # (A \cap {[-q,q]}^2) \leq C(K) q$ where $C(K)$ is a constant depending only on $K$. This extends and clarifies in the plane t…
We consider generalized Bochner-Riesz multipliers of the form $(1-ρ(ξ))_+^λ$ where $ρ(ξ)$ is the Minkowski functional of a convex domain in $\mathbb{R}^2$, with emphasis on domains for which the usual Carleson-Sjölin $L^p$ bounds can be improved. We produce convex domains for which previous results due to Seeger and Ziesler are not sharp. We identify two key properties of convex domains that lead to improved $L^p$ bounds for the associated Bochner-Riesz operators. First, we introduce the notion of the "additive energy" of the boundary of a convex domain. Second, we associate a set of direction…
Let $\mathcal{B}$ be a compact convex planar domain with smooth boundary of finite type and $\mathcal{B}_θ$ its rotation by an angle $θ$. We prove that for almost every $θ\in[0, 2π]$ the remainder $P_{\mathcal{B}_θ}(t)$ is of order $O_θ(t^{2/3-ζ})$ with a positive number $ζ$ independent of the domain.
We consider the Euler system set on a bounded convex planar domain, endowed with impermeability boundary conditions. This system is a model for the barotropic mode of the Primitive Equations on a rectangular domain. We show the existence of weak solutions with L^p vorticity for 4/3<= p <= 2, extending and enriching a previous result of Taylor. In the physically interesting case of a rectangular domain, a similar result holds for all 2<p<\infty as well. Moreover, we show the uniqueness of solutions with bounded initial vorticity. The main tool is a new BMO regularity estimate for the Dirichlet …
The covariogram g_K of a convex body K in E^d is the function which associates to each x in E^d the volume of the intersection of K with K+x. In 1986 G. Matheron conjectured that for d=2 the covariogram g_K determines K within the class of all planar convex bodies, up to translations and reflections in a point. This problem is equivalent to some problems in stochastic geometry and probability as well as to a particular case of the phase retrieval problem in Fourier analysis. It is also relevant for the inverse problem of determining the atomic structure of a quasicrystal from its X-ray diffrac…
Rigidity in the Planar Ulam Floating Body Problem with perimetral density $σ=\tfrac16$
研究平面凸域在周长密度σ=1/6下的乌拉姆浮体问题，利用Zindler旋转装置框架，将问题转化为与内接正六边形相关的二维动力系统。主要结果表明，圆盘是唯一在该周长密度下在任意位置都能保持平衡的凸域，为有理周长密度的凸设定提供了新的刚性结果。
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