Rational Lattice Research Articles

Improved hardness results for unique shortest vector problem

The unique shortest vector problem on a rational lattice is the problem of finding the shortest non-zero vector under the promise that it is unique (up to multiplication by −1). We give several incremental improvements on the known hardness of the unique shortest vector problem (uSVP) using standard techniques. This includes a deterministic reduction from the shortest vector problem to the uSVP, the NP-hardness of uSVP on (1+1poly(n))-unique lattices, and a proof that the decision version of uSVP defined by Cai [4] is in co-NP for n1/4-unique lattices.

On the Chromatic Numbers of Integer and Rational Lattices

In this paper, we give new upper bounds for the chromatic numbers for integer lattices and some rational spaces and other lattices. In particular, we have proved that for any concrete integer number d, the chromatic number of ℤn with critical distance \( \sqrt{2d} \) has a polynomial growth in n with exponent less than or equal to d (sometimes this estimate is sharp). The same statement is true not only in the Euclidean norm, but also in any lp norm. Moreover, we have given concrete estimates for some small dimensions as well as upper bounds for the chromatic number of ℚpn, where by ℚp we mean the ring of all rational numbers having denominators not divisible by some prime numbers.

Completeness of Gabor systems

We investigate the completeness of Gabor systems with respect to several classes of window functions on rational lattices. Our main results show that the time–frequency shifts of every finite linear combination of Hermite functions with respect to a rational lattice are complete in L2(R), thus generalizing a remark of von Neumann (and proved by Bargmann, Perelomov et al.). An analogous result is proven for functions that factor into certain rational functions and the Gaussian. The results are also interesting from a conceptual point of view since they show a vast difference between the completeness and the frame property of a Gabor system. In the terminology of physics we prove new results about the completeness of coherent state subsystems.

CONVEX CURVES AND A POISSON IMITATION OF LATTICES

We solve a randomized version of the following open question: is there a strictly convex, bounded curve such that the number of rational points on , with denominator , approaches infinity with ? Although this natural problem appears to be out of reach using current methods, we consider a probabilistic analogue using a spatial Poisson process that simulates the refined rational lattice , which we call , for each natural number . The main result here is that with probability 1 there exists a strictly convex, bounded curve such that as tends to infinity. The methods include the notion of a generalized affine length of a convex curve as defined by F. V. Petrov [Estimates for the number of rational points on convex curves and surfaces. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 344 (2007), 174–189; Engl. transl. J. Math. Sci. 147(6) (2007), 7218–7226].

Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices

We study various aspects of the dynamics induced by integer matrices on the invariant rational lattices of the torus in dimension 2 and greater. Firstly, we investigate the orbit structure when the toral endomorphism is not invertible on the lattice, characterising the pretails of eventually periodic orbits. Next we study the nature of the symmetries and reversing symmetries of toral automorphisms on a given lattice, which has particular relevance to (quantum) cat maps.

Distribution of periodic orbits for the Casati–Prosen map on rational lattices

We study numerically the periodic orbits of the Casati–Prosen map, a two-parameter reversible map of the torus, with zero entropy. For rational parameter values, this map preserves rational lattices, and each lattice decomposes into periodic orbits. We consider the distribution function of the periods over prime lattices, and its dependence on the parameters of the map. Based on extensive numerical evidence, we conjecture that, asymptotically, almost all orbits are symmetric, and that for a set of rational parameters having full density, the distribution function approaches the gamma-distribution R ( x ) = 1 − e − x ( 1 + x ) . These properties, which have been proved to hold for random reversible maps, were previously thought to require a stronger form of deterministic randomness, such as that displayed by rational automorphisms over finite fields. Furthermore, we show that the gamma-distribution is the limit of a sequence of singular distributions which are observed on certain lines in parameter space. Our experiments reveal that the convergence rate to R is highly non-uniform in parameter space, being slowest in sharply-defined regions reminiscent of resonant zones in Hamiltonian perturbation theory.

Constructing super Gabor frames: the rational time-frequency lattice case

For a time-frequency lattice Λ = Aℤ d × Bℤ d , it is known that an orthonormal super Gabor frame of length L exists with respect to this lattice if and only if |det(AB)| = 1/L. The proof of this result involves various techniques from multi-lattice tiling and operator algebra theory, and it is far from constructive. In this paper we present a very general scheme for constructing super Gabor frames for the rational lattice case. Our method is based on partitioning an arbitrary fundamental domain of the lattice in the frequency domain such that each subset in the partition tiles ℝ d by the lattice in the time domain. This approach not only provides us a simple algorithm of constructing various kinds of orthonormal super Gabor frames with flexible length and multiplicity, but also allows us to construct super Gabor (non-Riesz) frames with high order smoothness and regularity. Several examples are also presented.

Midwest cousins of Barnes–Wall lattices

Given a rational lattice and suitable set of linear transformations, we construct a cousin lattice. Sufficient conditions are given for integrality, evenness and unimodularity. When the input is a Barnes–Wall lattice, we get multi-parameter series of cousins. There is a subseries consisting of unimodular lattices which have ranks 2 d − 1 ± 2 d − k − 1 , for odd integers d ⩾ 3 and integers k = 1 , 2 , … , d − 1 2 . Their minimum norms are moderately high: 2 ⌊ d 2 ⌋ − 1 .

Similarity versus coincidence rotations of lattices

Abstract The groups of similarity and coincidence rotations of an arbitrary lattice Γ in d-dimensional Euclidean space are considered. It is shown that the group of similarity rotations contains the coincidence rotations as a normal subgroup. Furthermore, the structure of the corresponding factor group is examined. If the dimension d is a prime number, this factor group is an elementary Abelian d-group. Moreover, if Γ is a rational lattice, the factor group is trivial (d odd) or an elementary Abelian 2-group (d even).

Coincidence sites for rational lattices

The problem of identifying the coincidence site lattices produced by the disorientations of any rational lattice is investigated. A very simple new procedure for computing the CSLs is presented, based on Grimmer's reciprocity theorem and application of the reduction algorithm introduced by Buerger for finding reduced cells. The point symmetries of a lattice imply that any orientation relationship that produces a CSL can be alternatively expressed by a variety of different rotations. Two simple numerical methods (one based on integral matrix operations and one based on quaternions) are demonstrated for finding all the equivalent rotations that produce the same CSL and for obtaining a single “canonical” rotation as a representative of an equivalence class.

Variation of the number of lattice points in large balls

For rational lattices in Rd, d ‚ 4 we prove lower bounds for the variation of the number of lattice points inside balls of large radius with a variable centre. The obtained estimates are sharp for d ‚ 5.

Frames and the Feichtinger conjecture

We show that the conjectured generalization of the Bourgain-Tzafriri restricted-invertibility theorem is equivalent to the conjecture of Feichtinger, stating that every bounded frame can be written as a finite union of Riesz basic sequences. We prove that any bounded frame can at least be written as a finite union of linearly independent sequences. We further show that the two conjectures are implied by the paving conjecture. Finally, we show that Weyl-Heisenberg frames over rational lattices are finite unions of Riesz basic sequences.

A construction of admissible [formula omitted]-modules of level [formula omitted

By using generalized vertex algebras associated to rational lattices, we construct explicitly the admissible modules for the affine Lie algebra A 1 ( 1 ) of level - 4 3 . As an application, we show that the W ( 2 , 5 ) algebra with central charge c = - 7 investigated in (J. Algebra 270 (2003) 115) is a subalgebra of the simple vertex operator algebra L ( - 4 3 Λ 0 ) .

Adelic version of Margulis arithmeticity theorem

We generalize Margulis's S-arithmeticity theorem to the case when S can be taken as an infinite set of primes. Let R be the set of all primes including infinite one \(\infty\) and set \({mathbb Q}_{\infty}={\mathbb R}\). Let S be any subset of R. For each \(p\in S\), let \(G_p\) be a connected semisimple adjoint \({mathbb Q}_p\)-group and \(D_p\subset G_p({\mathbb Q}_p)\) be a compact open subgroup for each finite prime \(p\in S\). Let \((G_S, D_p)\) denote the restricted topological product of \(G_p({\mathbb Q}_{p})\)'s, \(p\in S\) with respect to \(D_p\)'s. Note that if S is finite, \((G_S, D_p)=\prod_{p\in S}G_p({\mathbb Q} _{p})\). We show that if \(\sum_{p\in S}\text{rank}_{ {\mathbb Q}_{p}}(G_p)\geq 2\), any irreducible lattice in \((G_S, D_p)\) is a rational lattice. We also present a criterion on the collections \(G_p\) and \(D_p\) for \((G_S, D_p)\) to admit an irreducible lattice. In addition, we describe discrete subgroups of \((G_{\mathbb A}, D_p)\) generated by lattices in a pair of opposite horospherical subgroups.

Number-theoretic method for practical indexing of crystal directions

Abstract The paper presents new number-theoretic approaches to the solution of orientation problems in single crystal studies. A fast method for indexing any rational crystal lattice direction was developed to be a new 3D generalization of the Euclid algorithm. Based on the latter, a method for indexing directions in any crystal lattice was developed, where the practical problem is reduced to the search of the shortest primitive lattice vector within a certain angular error for the direction in analysis. A multipurpose personal computer system for testing both real and reciprocal lattices of familiar crystals is devised. The compact-in-memory software represents a principally new “bottomless” atlas of any crystal stereographic projections, where each point of the projection field is indexed within a desired accuracy.

On the unique shortest lattice vector problem

We show that the problem of deciding whether a given rational lattice L has a vector of length less than some given value r is NP-hard, even under the promise that L has exactly zero or one vector of length less than r.

Pseudo-symmetries of Anosov maps and spectral statistics

The statistics of the quantum eigenvalues of certain families of nonlinear maps on the 2-torus are found not to belong to the universality classes one would expect from the symmetries of the (classical) dynamics the maps generate. These anomalies are shown to be caused by arithmetical quantum symmetries which do not have a classical limit. They are related to the dynamics generated by associated linear torus maps on particular rational lattices that form the support of the quantum Wigner functions.

Abelian Intertwining Algebras and Modules Related to Rational Lattices

It is proved that the abelian intertwining algebra constructed from a rational lattice is rational with respect to certain data. A related conjecture of Huang on rational vertex operator algebras is proved as well.

On orthogonal bases for rational lattices

On orthogonal bases for rational lattices

Finite quaternionic matrix groups

Let D \mathcal {D} be a definite quaternion algebra such that its center has degree d d over Q \mathbb {Q} . A subgroup G G of G L n ( D ) GL_n(\mathcal {D}) is absolutely irreducible if the Q \mathbb {Q} -algebra spanned by the matrices in G G is D n × n \mathcal {D}^{n\times n} . The finite absolutely irreducible subgroups of G L n ( D ) GL_n(\mathcal {D}) are classified for n d ≤ 10 nd \leq 10 by constructing representatives of the conjugacy classes of the maximal finite ones. Methods to construct the groups and to deal with the quaternion algebras are developed. The investigation of the invariant rational lattices yields quaternionic structures for many interesting lattices.