Related papers: Kissing numbers and transference theorems from gen…
This is the third in a series of papers dealing with the algebraic theory of infinite classical lattices. This paper presents a theory of single measurements on a lattice which we represent as comprising a finite subvolume--the system of…
We derive two lower bounds on entanglement of formation for arbitrary mixed Gaussian states by two distinct methods. To achieve the first one we use a local measurement procedure derived by Giedke et al [Quantum Inf. and Comp. vol.1, 79…
Let $E(x;\omega)$ be the error term for the number of integer lattice points lying inside a $3$-dimensional Cygan-Kor\'anyi spherical shell of inner radius $x$ and gap width $\omega(x)>0$. Assuming that $\omega(x)\to0$ as $x\to\infty$, and…
We study transfer principles for upper bounds of motivic exponential functions and for linear combinations of such functions, directly generalizing the transfer principles from [7] by Cluckers-Loeser and [13, Appendix B] by Shin-Templier…
L\'{a}szl\'{o} Fejes T\'{o}th and Alad\'{a}r Heppes proposed the following generalization of the kissing number problem. Given a ball in $\mathbb{R}^d$, consider a family of balls touching it, and another family of balls touching the first…
How many unit $n-$dimensional spheres can simultaneously touch or kiss a central $n-$dimensional unit sphere? Beyond mathematics this question has implications for fields such as cryptography and the structure of biologic and chemical…
Banaszczyk's inequality establishes a tail estimate for the discrete Gaussian measure on a lattice in $\mathbb{R}^n$. This classic result has been influential and plays an important role in lattice-based cryptography. An improvement of the…
We study the height and width of a Galton--Watson tree with offspring distribution B satisfying E(B)=1, 0 < Var(B) < infinity, conditioned on having exactly n nodes. Under this conditioning, we derive sub-Gaussian tail bounds for both the…
We revisit the problem of influencing the sex ratio of a population by subjecting reproduction of each family to some stopping rule. As an easy consequence of the strong law of large numbers, no such modification is possible in the sense…
We estimate the distribution of relatively $r$-prime lattice points in number fields $K$ with their components having a norm less than $x$. In the previous paper we obtained uniform upper bounds as $K$ runs through all number fields under…
This paper studies the lattice agreement problem and proposes a stronger form, $\varepsilon$-bounded lattice agreement, that enforces an additional tightness constraint on the outputs. To formalize the concept, we define a quasi-metric on…
In statistical learning theory, a generalization bound usually involves a complexity measure imposed by the considered theoretical framework. This limits the scope of such bounds, as other forms of capacity measures or regularizations are…
We consider a weighted lattice $Z^d$ with conductance $\mu_e=|e|^{-\alpha}$. We show that the heat kernel of a variable speed random walk on it satisfies a two-sided Gaussian bound by using an intrinsic metric. We also show that when $d=2$…
We introduce a "microcanonical" measure (complying with the "general canonical principle") over the second moments of pure bosonic Gaussian states under an energy constraint. We determine the average fidelity for the teleportation of states…
An information measure based on fractional partitions of a set is used to derive a general dependence balance inequality for communication. This inequality is used to obtain new upper bounds on reliable and secret rates for multiterminal…
We prove an upper bound on the expected $\ell_p$ injective norm of sums of subgaussian random tensors. Our proof is simple and does not rely on any explicit geometric or chaining arguments. Instead, it follows from a simple application of…
We present a general approach, based on exponential inequalities, to derive bounds on the generalization error of randomized learning algorithms. Using this approach, we provide bounds on the average generalization error as well as bounds…
We survey problems and results from combinatorial geometry in normed spaces, concentrating on problems that involve distances. These include various properties of unit-distance graphs, minimum-distance graphs, diameter graphs, as well as…
Comparing recent lattice results on chiral fermions and old continuum results for the index puzzling questions arise. To clarify this issue we start with a critical reconsideration of the results on finite lattices. We then work out various…
New bounds for the $k$-th order derivatives of the solutions of the normal and multivariate normal Stein equations are obtained. Our general order bounds involve fewer derivatives of the test function than those in the existing literature.…