Related papers: Weak admissibility, primitivity, o-minimality, and…
In development of the started activity on lattice analogues of $W$-algebras, we define the notion of lattice $W_{\infty}$-algebra, accociated with lattice integrable system with infinite set of fields. Various kinds of reduction to lattice…
We prove an estimate for the number of lattice points lying in certain non-convex Euclidean domains of interest in Diophantine approximation. As an application, we generalise a result of Kruse (1964) concerning the almost sure order of…
We study the statistical properties of the counting function of lattice points inside thin annuli. By a conjecture of Bleher and Lebowitz, if the width shrinks to zero, but the area converges to infinity, the distribution converges to the…
Via multilinear algebra, we formulate a criterion for connectedness in the parametric geometry of numbers in terms of pencils, which are certain algebraic varieties in the space of matrices. As a consequence, we obtain a connectedness…
The theory of stochastic approximations form the theoretical foundation for studying convergence properties of many popular recursive learning algorithms in statistics, machine learning and statistical physics. Large deviations for…
Recently in joint work with E. Sert, we proved sharp boundedness results on discrete fractional integral operators along binary quadratic forms. Present work vastly enhances the scope of those results by extending boundedness to bivariate…
For a tuple $(\theta_1,..,\theta_M)$ of complex number, buliding on the approximation techniques in earlier papers of this series, this paper engages in deducing lower estimates on the transcendence degree of the field generated by…
We propose a numerical integrator for determining low-rank approximations to solutions of large-scale matrix differential equations. The considered differential equations are semilinear and stiff. Our method consists of first splitting the…
We revisit Kapranov and Voevodsky's idea of spaces modelled on combinatorial pasting diagrams, now as a framework for higher-dimensional rewriting and the basis of a model of weak omega-categories. In the first part, we elaborate on…
We prove new quantitative Schmidt-type theorem for Diophantine approximations with restraint denominators on fractals (more precisely, on $M_0$-sets). Our theorems introduce a sharp balance condition between the growth rate of the sequence…
Approximate lattices are aperiodic generalisations of lattices of locally compact groups that were first studied in seminal work of Yves Meyer. They are defined as those uniformly discrete approximate subgroups (symmetric subsets stable…
Under the assumption that the approximating function $\psi$ is monotonic, the classical Khintchine-Groshev theorem provides an elegant probabilistic criterion for the Lebesgue measure of the set of $\psi$-approximable matrices in $\R^{mn}$.…
There is a rich theory of existence theorems for minimizers over reflexive Sobolev spaces (ex. Eberlein-\v{S}mulian theorem). However, the existence theorems for many variational problems over non-reflexive Sobolev spaces remain…
Gallagher's theorem describes the multiplicative diophantine approximation rate of a typical vector. We establish a fully-inhomogeneous version of Gallagher's theorem, a diophantine fibre refinement, and a sharp and unexpected threshold for…
The present paper establishes upper and lower bounds on the speed of approximation in a wide range of natural Diophantine approximation problems. The upper and lower bounds coincide in many cases, giving rise to optimal results in…
The paper provides new upper and lower bounds for the multivariate Laplace approximation under weak local assumptions. Their range of validity is also given. An application to an integral arising in the extension of the Dixon's identity is…
We reduce the principal problem of Additive Number Theory of whether an infinite sequence of integers constitutes a finite basis for the integers to a Diophantine problem involving the difference set of the sequence, by proving a formula…
We present a class of lattices in R^d (d >= 2) which we call GL-lattices and conjecture that any lattice is such. This conjecture is referred to as GLC. Littlewood's conjecture amounts to saying that Z^2 is GL. We then prove existence of GL…
We study point sets arising from cut-and-project constructions. An important class is weak model sets, which include squarefree numbers and visible lattice points. For such model sets, we give a non-trivial upper bound on their pattern…
Lattice rounding in Euclidean space can be viewed as finding the nearest point in the orbit of an action by a discrete group, relative to the norm inherited from the ambient space. Using this point of view, we initiate the study of…