Restriction of 3D arithmetic Laplace eigenfunctions to a plane

For arbitrary real b > 1 and for a large real parameter t let R (b, t) be the number of lattice points between the curve by = x (1 ≤x≤t) and the x-axis, the points on the x-axis being counted with the factor 1 / 2. Furthermore, let the lattice rest\( {\mit\Gamma} (b, t) \) be defined as the difference between R (b, t) and the area of the domain. The asymptotic behaviour of \( {\mit\Gamma} (b, t) \) for \( t\rightarrow \infty \) and also for b→ 1+ are investigated.

Higher moments for lattice point discrepancy of convex domains and annuli

Motivated by long-standing conjectures on the discretization of classical inequalities in the Geometry of Numbers, we investigate a new set of parameters, which we call \emph{packing minima}, associated to a convex body $K$ and a lattice $\Lambda$. These numbers interpolate between the successive minima of $K$ and the inverse of the successive minima of the polar body of $K$, and can be understood as packing counterparts to the covering minima of Kannan & Lov\'{a}sz (1988). As our main results, we prove sharp inequalities that relate the volume and the number of lattice points in $K$ to the sequence of packing minima. Moreover, we extend classical transference bounds and discuss a natural class of examples in detail.

Rank-1 lattice rules based on a weighted star discrepancy with weights of a product form have been previously constructed under the assumption that the number of points is prime. Here, we extend these results to the non-prime case. We show that if the weights are summable, there exist lattice rules whose weighted star discrepancy is O(n −1+δ ), for any δ > 0, with the implied constant independent of the dimension and the number of lattice points, but dependent on δ and the weights. Then we show that the generating vector of such a rule can be constructed using a component-by-component (CBC) technique. The cost of the CBC construction is analysed in the final part of the paper.

This paper considers the local integrability condition for generalised translation-invariant systems and its relation to the Calderon integrability condition, the temperateness condition and the uniform counting estimate. It is shown that sufficient and necessary conditions for satisfying the local integrability condition are closely related to lower and upper bounds on the number of lattice points that intersect with the translates of a compact set. The results are complemented by examples that illustrate the crucial interplay between the translation subgroups and the generating functions of the system.

On the number of lattice points in convex symmetric bodies and their duals

A new estimate is obtained for the residue R in the asymptotics of the number of integer points in a ball of radius a. The estimate has the form R ≪ a 17/14 + ɛ .

In their survey article on cyclic Steiner Quadruple Systems SQS(v) M. J. Grannel and T. S. Griggs advanced the conjecture (cf. [8, p. 412]) that their necessary condition for the existence of S-cyclic SQS(v) (cf. [7, p. 51]) is also sufficient. Some years prior to that E. Kohler [10] used a graph theoretical method to construct S-cyclic SQS(v). This method was extended in [17]-[20] and eventually used to reduce the conjecture of Grannel and Griggs to a number theoretic claim (cf. also [21], research problem 146). The main purpose of the present paper is to attack this claim. For the long intervals we have to distinguish four cases. The proof of cases I–III can be accomplished by a thorough study of how the multiples of a certain set belonging to the first column of a certain matrix (the elements of which are essentially the vertices of a graph corresponding to SQS(2p)) are distributed over the columns. The proof is by contradiction. Case IV is most difficult to treat and could only be dealt with by very deep lying means. We have to use an asymptotic formula on the number of lattice points (x,y) with xy ≡ 1 mod p (we speak of 1-points) in a rectangle and this formula shows that the 1-points are equidistributed. But even so our claim could not be proved for all intervals of admissible length. Intervals [a,b] with ^p—_m+1 <a <b < ^p−_m for some m and 3√4 p <b−a <p ^11−15 could not be covered. In the last section we discuss some conclusions which would follow from the non-existence of complete intervals.

Analogs of classical results on accumulated spectrograms, the sum of spectrograms of eigenfunctions of localization operators, are established for Gabor multipliers. We show that the lattice ℓ1 distance between the accumulated spectrogram and the indicator function of the Gabor multiplier mask is bounded by the number of lattice points near the boundary of the mask and that this bound is sharp in general. The methods developed for the proofs are also used to show that the Weyl-Heisenberg ensemble restricted to a lattice is hyperuniform when the Gabor frame is tight.

We present a general framework for computing various optimal embeddings of undirected and directed connected graphs in two and multi-dimensional integer lattices in time sub-exponential either in the minimum number n of lattice points used by such optimal embeddings or in the budget upper bound b on the number of lattice points that may be used in an embedding. The sub-exponential upper bounds in the two dimensional case and d-dimensional case are respectively of the form 2\(^{O(\sqrt{ln}log n)}\), 2\(^{O(\sqrt{lb}log b)}\) and 2\(^{O(dl^{1/d_n (d-1)/d}{\rm log} n)}\), 2\(^{O(dl^{1/d_b (d-1)/d}{\rm log} b)}\), where l stands for the degree of the allowed overlap. For the problem of minimum total edge length planar or multi-dimensional embedding or layout of a graph and the problem of an optimal protein folding in the so called HP model we obtain the upper bounds in terms of n. Note that in case of protein folding n is also the size of the input. The list of problems for which we can derive the upper bounds in terms of b includes among other things:

This paper applies error-exponent and dispersionstyle analyses to derive finite-blocklength achievability bounds for low-density parity-check (LDPC) codes over the point-to-point channel (PPC) and multiple access channel (MAC). The error-exponent analysis applies Gallager's error exponent to bound achievable symmetrical and asymmetrical rates in the MAC. The dispersion-style analysis begins with a generalization of the random coding union (RCU) bound from random code ensembles with i.i.d. codewords to random code ensembles in which codewords may be statistically dependent; this generalization is useful since the codewords of random linear codes such as LDPC codes are dependent. Application of the RCU bound yields finite-blocklength error bounds and asymptotic achievability results for both i.i.d. random codes and LDPC codes. For discrete, memoryless channels, these results show that LDPC codes achieve first- and second-order performance that is optimal for the PPC and identical to the best prior results for the MAC.

This paper establishes expectation and variance asymptotics for statistics of the Poisson–Voronoi approximation of general sets, as the underlying intensity of the Poisson point process tends to infinity. Statistics of interest include volume, surface area, Hausdorff measure, and the number of faces of lower-dimensional skeletons. We also consider the complexity of the so-called Voronoi zone and the iterated Voronoi approximation. Our results are consequences of general limit theorems proved with an abstract Steiner-type formula applicable in the setting of sums of stabilizing functionals.

In this paper, we present a low-complexity list sphere search algorithm for achieving near-optimal a posteriori probability (APP) detection in iterative detection and decoding (IDD). Motivated by the fact that the list sphere decoding searching a fixed number of lattice points is inefficient in many scenarios, we design a criterion to search lattice points with non-vanishing likelihood and derive the optimal sphere radius satisfying this requirement. Further, in order to exploit the sphere constraint as it is instead of using necessary conditioned versions, we incorporate a probabilistic tree pruning strategy into the list sphere search. Through simulations on realistic IDD systems, we show that the proposed method provides considerable complexity savings while maintaining near-optimal performance.

This paper takes a step towards theoretical analysis of the relationship between word embeddings and context embeddings in models such as word2vec. We start from basic probabilistic assumptions on the nature of word vectors, context vectors, and text generation. These assumptions are supported either empirically or theoretically by the existing literature. Next, we show that under these assumptions the widely-used word-word PMI matrix is approximately a random symmetric Gaussian ensemble. This, in turn, implies that context vectors are reflections of word vectors in approximately half the dimensions. As a direct application of our result, we suggest a theoretically grounded way of tying weights in the SGNS model.

We consider the phase retrieval problem, which asks to recover a structured n-dimensional signal from m quadratic measurements. In many imaging contexts, it is beneficial to enforce a sparsity prior on the signal to reduce the number of measurements necessary for recovery. However, the best known methodologies for sparse phase retrieval have a sub-optimal quadratic dependency on the sparsity level of the signal at hand. In this work, we instead model signals as living in the range of a deep generative neural network G : ℝk → ℝn. We show that under the model of a d-layer feed forward neural network with Gaussian weights, m = O(kd2 log n) generic measurements suffice for the l 2 empirical risk minimization problem to have favorable geometry. In particular, we exhibit a descent direction for all points outside of two arbitrarily small neighborhoods of the true k-dimensional latent code and a negative reflection of it. Our proof is based on showing the sufficiency of two deterministic conditions on the generator and measurement matrices, which are satisfied with high probability under random Gaussian ensembles. We corroborate these results with numerical experiments showing that enforcing a generative prior via empirical risk minimization outperforms sparse phase retrieval methods.