Admissible Lattices Research Articles

SL2 representations and relative property (T)

We investigate relative property (T) of Kazhdan-Margulis for (SL2(R)⋉Rn,Rn), mainly where R is a discrete finitely generated commutative ring. The action of SL2(R) on Rn is defined through admissible lattices of the irreducible representations of sl2(C). Both positive and negative results are obtained. Applications of these results will be given, including the establishment of property (T) for certain Kac-Moody type groups over commutative rings.

On unique recovery of finite-valued integer signals and admissible lattices of sparse hypercubes

The paper considers the problem of unique recovery of sparse finite-valued integer signals using a single linear integer measurement. For l-sparse signals in {mathbb {Z}}^n, 2l<n, with absolute entries bounded by r, we construct an 1times n measurement matrix with maximum absolute entry Delta =O(r^{2l-1}). Here the implicit constant depends on l and n and the exponent 2l-1 is optimal. Additionally, we show that, in the above setting, a single measurement can be replaced by several measurements with absolute entries sub-linear in Delta. The proofs make use of results on admissible (n-1)-dimensional integer lattices for m-sparse n-cubes that are of independent interest.

Counting lattice points and weak admissibility of a lattice and its dual

We prove a counting theorem concerning the number of lattice points for the dual lattices of weakly admissible lattices in an inhomogeneously expanding box. The error term is expressed in terms of a certain function ν(Γ⊥, ·) of the dual lattice Γ⊥, and we carefully analyse the relation of this quantity with ν(Γ, ·). In particular, we show that ν(Γ⊥, ·) = ν(Γ, ·) for any unimodular lattice of rank 2, but that for higher ranks it is in general not possible to bound one function in terms of the other. This result relies on Beresnevich’s recent breakthrough on Davenport’s problem regarding badly approximable points on submanifolds of ℝn. Finally, we apply our counting theorem to establish asymptotics for the number of Diophantine approximations with bounded denominator as the denominator bound gets large.

Numerical Performance of Optimized Frolov Lattices in Tensor Product Reproducing Kernel Sobolev Spaces

In this paper, we deal with several aspects of the universal Frolov cubature method, which is known to achieve optimal asymptotic convergence rates in a broad range of function spaces. Even though every admissible lattice has this favorable asymptotic behavior, there are significant differences concerning the precise numerical behavior of the worst-case error. To this end, we propose new generating polynomials that promise a significant reduction in the integration error compared to the classical polynomials. Moreover, we develop a new algorithm to enumerate the Frolov points from non-orthogonal lattices for numerical cubature in the d-dimensional unit cube [0,1]^d. Finally, we study Sobolev spaces with anisotropic mixed smoothness and compact support in [0,1]^d and derive explicit formulas for their reproducing kernels. This allows for the simulation of exact worst-case errors which numerically validate our theoretical results.

On a Counting Theorem for Weakly Admissible Lattices

Abstract We give a precise estimate for the number of lattice points in certain bounded subsets of $\mathbb{R}^{n}$ that involve “hyperbolic spikes” and occur naturally in multiplicative Diophantine approximation. We use Wilkie’s o-minimal structure $\mathbb{R}_{\exp }$ and expansions thereof to formulate our counting result in a general setting. We give two different applications of our counting result. The 1st one establishes nearly sharp upper bounds for sums of reciprocals of fractional parts and thereby sheds light on a question raised by Lê and Vaaler, extending previous work of Widmer and of the author. The 2nd application establishes new examples of linear subspaces of Khintchine type thereby refining a theorem by Huang and Liu. For the proof of our counting result, we develop a sophisticated partition method that is crucial for further upcoming work on sums of reciprocals of fractional parts over distorted boxes.

Sums of reciprocals of fractional parts

Let [Formula: see text] and [Formula: see text]. We consider the sum [Formula: see text]. Sharp upper bounds are known when [Formula: see text], using continued fractions or the three-distance theorem. However, these techniques do not seem to apply in higher dimension. We introduce a different approach, based on a general counting result of Widmer for weakly admissible lattices, to establish sharp upper bounds for arbitrary [Formula: see text]. Our result also sheds light on a question raised by Lê and Vaaler in 2013 on the sharpness of their lower bound [Formula: see text].

A note on the dispersion of admissible lattices

In this note we show that the volume of axis-parallel boxes in Rd which do not intersect an admissible lattice L⊂Rd is uniformly bounded. In particular, this implies that the dispersion of the dilated lattices N−1∕dL restricted to the unit cube is of the (optimal) order N−1 as N goes to infinity. This result was obtained independently by Temlyakov (2017).

Digital net properties of a polynomial analogue of Frolov's construction

Frolov's cubature formula on the unit hypercube has been considered important since it attains an optimal rate of convergence for various function spaces. Its integration nodes are given by shrinking a suitable full rank Z-lattice in Rd and taking all points inside the unit cube. The main drawback of these nodes is that they are hard to find computationally, especially in high dimensions. In such situations, quasi-Monte Carlo (QMC) rules based on digital nets have proven to be successful. However, there is still no construction known that leads to QMC rules which are optimal in the same generality as Frolov's.In this paper we investigate a polynomial analog of Frolov's cubature formula, which we expect to be important in this respect. This analog is defined in a field of Laurent series with coefficients in a finite field. A similar approach was previously studied in [M. B. Levin. Adelic constructions of low discrepancy sequences. Online Journal of Analytic Combinatorics. Issue 5, 2010.].We show that our construction is a (t,m,d)-net, which also implies bounds on its star-discrepancy and the error of the corresponding cubature rule. Moreover, we show that our cubature rule is a QMC rule, whereas Frolov's is not, and provide an algorithm to determine its integration nodes explicitly.To this end we need to extend the notion of (t,m,d)-nets to fit the situation that the points can have infinite digit expansion and develop a duality theory. Additionally, we adapt the notion of admissible lattices to our setting and prove its significance.

Canonical tilting relative generators

Given a relatively projective birational morphism f:X→Y of smooth algebraic spaces with dimension of fibers bounded by 1, we construct tilting relative (over Y) generators TX,f and SX,f in Db(X). We develop a piece of general theory of strict admissible lattice filtrations in triangulated categories and show that Db(X) has such a filtration L where the lattice is the set of all birational decompositions f:X→gZ→hY with smooth Z. The t-structures related to TX,f and SX,f are proved to be glued via filtrations left and right dual to L. We realise all such Z as the fine moduli spaces of simple quotients of OX in the heart of the t-structure for which SX,g is a relative projective generator over Y. This implements the program of interpreting relevant smooth contractions of X in terms of a suitable system of t-structures on Db(X).

On the orthogonality of the Chebyshev–Frolov lattice and applications

We deal with lattices that are generated by the Vandermonde matrices associated to the roots of Chebyshev polynomials. If the dimension d of the lattice is a power of two, i.e. $$d=2^m, m \in \mathbb {N}$$ , the resulting lattice is an admissible lattice in the sense of Skriganov. We prove that these lattices are orthogonal and possess a lattice representation matrix with orthogonal columns and entries not larger than 2 (in modulus). In particular, we clarify the existence of orthogonal admissible lattices in higher dimensions. The orthogonality property allows for an efficient enumeration of these lattices in axis parallel boxes. Hence they serve for a practical implementation of the Frolov cubature formulas, which recently drew attention due to their optimal convergence rates in a broad range of Besov–Lizorkin–Triebel spaces. As an application, we efficiently enumerate the Frolov cubature nodes in the d-cube $$[-1/2,1/2]^d$$ up to dimension $$d=16$$ .

A fractional generalization of the classical lattice dynamics approach

We develop physically admissible lattice models in the harmonic approximation which define by Hamilton’s variational principle fractional Laplacian matrices of the forms of power law matrix functions on the n-dimensional periodic and infinite lattice in n=1,2,3,.. dimensions. The present model which is based on Hamilton’s variational principle is confined to conservative non-dissipative isolated systems. The present approach yields the discrete analogue of the continuous space fractional Laplacian kernel. As continuous fractional calculus generalizes differential operators such as the Laplacian to non-integer powers of Laplacian operators, the fractional lattice approach developed in this paper generalized difference operators such as second difference operators to their fractional (non-integer) powers. Whereas differential operators and difference operators constitute local operations, their fractional generalizations introduce nonlocal long-range features. This is true for discrete and continuous fractional operators. The nonlocality property of the lattice fractional Laplacian matrix allows to describe numerous anomalous transport phenomena such as anomalous fractional diffusion and random walks on lattices. We deduce explicit results for the fractional Laplacian matrix in 1D for finite periodic and infinite linear chains and their Riesz fractional derivative continuum limit kernels.The fractional lattice Laplacian matrix contains for α=2 the classical local lattice approach with well known continuum limit of classic local standard elasticity, and for other integer powers to gradient elasticity. We also present a generalization of the fractional Laplacian matrix to n-dimensional cubic periodic (nD tori) and infinite lattices. We show that in the continuum limit the fractional Laplacian matrix yields the well-known kernel of the Riesz fractional Laplacian derivative being the kernel of the fractional power of Laplacian operator. In this way we demonstrate the interlink of the fractional lattice approach with existing continuous fractional calculus. The developed approach appears to be useful to analyze fractional random walks on lattices as well as fractional wave propagation phenomena in lattices.

New Tools for Classifying Hamiltonian Circle Actions with Isolated Fixed Points

For every compact almost complex manifold $$(\mathsf {M},\mathsf {J})$$ ( M , J ) equipped with a $$\mathsf {J}$$ J -preserving circle action with isolated fixed points, a simple algebraic identity involving the first Chern class is derived. This enables us to construct an algorithm to obtain linear relations among the isotropy weights at the fixed points. Suppose that $$\mathsf {M}$$ M is symplectic and the action is Hamiltonian. If the manifold satisfies an extra so-called positivity condition, then this algorithm determines a family of vector spaces that contain the admissible lattices of weights. When the number of fixed points is minimal, this positivity condition is necessarily satisfied whenever $$\dim (\mathsf {M})\le 6$$ dim ( M ) ≤ 6 and, when $$\dim (\mathsf {M})=8$$ dim ( M ) = 8 , whenever the $$S^1$$ S 1 -action extends to an effective Hamiltonian $$T^2$$ T 2 -action, or none of the isotropy weights is $$1$$ 1 . Moreover, there are no known examples with a minimal number of fixed points contradicting this condition, and their existence is related to interesting questions regarding fake projective spaces. We run the algorithm for $$\dim (\mathsf {M})\le 8$$ dim ( M ) ≤ 8 , quickly obtaining all the possible families of isotropy weights. In particular, we simplify the proofs of Ahara and Tolman for $$\dim (\mathsf {M})=6$$ dim ( M ) = 6 and, when $$\dim (\mathsf {M})=8$$ dim ( M ) = 8 , we prove that the equivariant cohomology ring, Chern classes, and isotropy weights agree with those of $${\mathbb {C}}P^4$$ C P 4 with the standard $$S^1$$ S 1 -action (thereby proving the symplectic Petrie conjecture in this setting).

Resolution of the wavefront set via discrete sets

AbstractWe introduce discrete wave‐front sets of sup type and prove that they coincide with the Hörmander wave‐front set of a distribution. To that end we recall the notion of admissible lattice pairs and wave‐front sets in Fourier‐Lebesgue spaces. (© 2013 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Gabor pairs, and a discrete approach to wave-front sets

We introduce admissible lattices and Gabor pairs to define discrete versions of wave-front sets with respect to Fourier–Lebesgue and modulation spaces. We prove that these wave-front sets agree with each other and with corresponding wave-front sets of “continuous type”. This implies that the coefficients of a Gabor frame expansion of f are parameter dependent, and describe the wave-front set of f.

On low discrepancy sequences and low discrepancy ergodic transformations of the multidimensional unit cube

In this paper we describe a third class of low discrepancy sequences. Using a lattice Γ ⊂ ℝs, we construct Kronecker-like and van der Corput-like ergodic transformations T1,Γ and T2,Γ of [0, 1)s. We prove that for admissible lattices Γ, (Tν,Γn(x))n≥0 is a low discrepancy sequence for all x ∈ [0, 1)s and ν ∈ {1, 2}. We also prove that for an arbitrary polyhedron P ⊂ [0, 1)s, for almost all lattices Γ ∈ Ls = SL(s,ℝ)/SL(s, ℤ) (in the sense of the invariant measure on Ls), the following asymptotic formula $$\# \{ 0 \le n < N:T_{v,\Gamma }^n(x) \in P\} = NvolP + O({(\ln N)^{s + \varepsilon }}),N \to \infty$$ holds with arbitrary small ɛ > 0, for all x ∈ [0, 1)s, and ν ∈ {1, 2}.

Factorization symmetry in the lattice Boltzmann method

A non-perturbative algebraic theory of lattice Boltzmann method is developed based on a symmetry of a product. It involves three steps: (i) Derivation of admissible lattices in one spatial dimension through a matching condition which imposes restricted extension of higher-order Gaussian moments, (ii) Special quasi-equilibrium distribution function found analytically in closed form on the product-lattice in two and three spatial dimensions, and which proves factorization of quasi-equilibrium moments, and (iii) Algebraic method of pruning based on a one-into-one relation between groups of discrete velocities and moments. Two routes of constructing lattice Boltzmann equilibria are distinguished. Present theory includes previously known limiting and special cases of lattices, and enables automated derivation of lattice Boltzmann models from two-dimensional tables, by finding roots of one polynomial and solving a few linear systems.

Bee+Cl@k

We build on prior work on intra-array memory reuse, for which a general theoretical framework was proposed based on lattice theory. Intra-array memory reuse is a way of reducing the size of a temporary array by folding, thanks to affine mappings and modulo operations, reusing memory locations when they contain a value not used later. We describe the algorithms needed to implement such a strategy. Our implementation has two parts. The first part, Bee, uses the source-to-source transformer ROSE to extract from the program all necessary information on the lifetime of array elements and to generate the code after memory reduction. The second part, Cl@k, is a stand-alone mathematical tool dedicated to optimizations on polyhedra, in particular the computation of successive minima and the computation of good admissible lattices, which are the basis for lattice-based memory reuse. Both tools are developed in C++ and use linear programming and polyhedra manipulations. They can be used either for embedded program optimizations, e.g., to limit memory expansion introduced for parallelization, or in high-level synthesis, e.g., to design memories between communicating hardware accelerators.

Wess-Zumino-Witten term on the lattice

We construct the Wess-Zumino-Witten (WZW) term in lattice gauge theory by using a Dirac operator which obeys the Ginsparg-Wilson relation. Topological properties of the WZW term known in the continuum are reproduced on the lattice as a consequence of a non-trivial topological structure of the space of admissible lattice gauge fields. In the course of this analysis, we observe that the gauge anomaly generally implies that there is no basis of a Weyl fermion which leads to a single-valued expectation value in the fermion sector. The lattice Witten term, which carries information of a gauge path along which the gauge anomaly is integrated, is separated from the WZW term and the multivaluedness of the Witten term is shown to be related to the homotopy group $\pi_{2n+1}(G)$. We also discuss the global $\SU(2)$ anomaly on the basis of the WZW term.

Representations of Chevalley Groups Arising from Admissible Lattices

Representations of Chevalley Groups Arising from Admissible Lattices

Representations of Chevalley groups arising from admissible lattices

The modules for a Chevalley group arising from admissible lattices in an irreducible module for the associated complex semisimple Lie algebra are studied. It is proved that the transpose of such a module is still in this collection and generically the cohomology modules of line bundles on the flag varieties are in this collection also. In the rank 1 case, all modules in this collection are indecomposable and we hope this is true in general.