Related papers: On a discrete John-type theorem

We study the convexity properties of the generalized trigonometric functions considered as functions of parameter. We show that $p\to\sin_p(y)$ and $p\to\cos_p(y)$ are log-concave on the appropriate intervals while $p\to\tan_p(y)$ is…

We prove a general combination theorem for discrete subgroups of $\mathrm{PGL}(n,\mathbb{R})$ preserving properly convex open subsets in the projective space $\mathbb{P}(\mathbb{R}^n)$, in the spirit of Klein and Maskit. We use it in…

Assuming the Generalized Riemann Hypothesis, we obtain a lower bound within a constant factor of the conjectured asymptotic result for the second moment for primes in an individual arithmetic progression in short intervals. Previous results…

We give a systematic and thorough study of geometric notions and results connected to Minkowski's measure of symmetry and the extension of the well-known Minkowski functional to arbitrary, not necessarily symmetric convex bodies K on any…

We consider two well-known problems: upper bounding the volume of lower dimensional ellipsoids contained in convex bodies given their John ellipsoid, and lower bounding the volume of ellipsoids containing projections of convex bodies given…

We derive an asymptotic formula for the divisor function $\tau(k)$ in an arithmetic progression $k\equiv a(\bmod \ q)$, uniformly for $q\leq X^{\Delta_{n,l}}$ with $(q,a)=1$. The parameter $\Delta_{n,l}$ is defined as $$…

We construct a generalization of the two-dimensional Wess-Zumino-Witten model on a $2n$-dimensional K\"ahler manifold as a group-valued non-linear sigma model with an anomaly term containing the K\"ahler form. The model is shown to have an…

Generalizing the notion of Newton polytope, we define the Newton-Okounkov body, respectively, for semigroups of integral points, graded algebras, and linear series on varieties. We prove that any semigroup in the lattice Z^n is…

This paper focuses on the metric properties of L\"uroth well approximable numbers, studying analogous of classical results, namely the Khintchine Theorem, the Jarn\'ik--Besicovitch Theorem, and the result of Dodson. A supplementary proof is…

The Euclidean algorithm makes possible a simple but powerful generalization of Taylor's theorem. Instead of expanding a function in a series around a single point, one spreads out the spectrum to include any number of points with given…

We prove that for a large class of multiplicative functions, referred to as generalized divisor functions, it is possible to find a lower bound for the corresponding variance in arithmetic progressions. As a main corollary, we deduce such a…

In this paper we develop a theory of convexity for a free Abelian group M (the lattice of integer points), which we call theory of discrete convexity. We characterize those subsets X of the group M that could be call "convex". One property…

The Jacobian elliptic functions are standard forms of elliptic functions, and they were independently introduced by C.G.J. Jacobi and N.H. Abel. In this paper, we study the unimodality of Taylor expansion coefficients of the Jacobian…

It is a classical result in rational approximation theory that certain non-smooth or singular functions, such as $|x|$ and $x^{1/p}$, can be efficiently approximated using rational functions with root-exponential convergence in terms of…

We adapt an argument of Tao and Vu to show that if $\lambda_1\le\cdots\le\lambda_d$ are the successive minima of an origin-symmetric convex body $K$ with respect to some lattice $\Lambda<\mathbb{R}^d$, and if we set…

Let $G$ be a multiplicative subgroup of the prime field $\mathbb F_p$ of size $|G|> p^{1-\kappa}$ and $r$ an arbitrarily fixed positive integer. Assuming $\kappa=\kappa(r)>0$ and $p$ large enough, it is shown that any proportional subset…

We generalise the Fundamental Theorem of Calculus to higher dimensions. Our generalisation is based on the observation that the antiderivative of a function of $n$-variables is a solution of a partial differential equation of order $n$…

In this article we discuss classical theorems from Convex Geometry in the context of topological drawings and beyond. In a simple topological drawing of the complete graph $K_n$, any two edges share at most one point: either a common vertex…

Let $A \in \{ \mathbb{Z}, \mathbb{R} \}$ and $X \subset \mathbb{R}^d$ be a bounded set. Affine transformations given by an automorphism of $\mathbb{Z}^d$ and a translation in $A^d$ are called (affine) $A$-unimodular transformations. The…

One of the most fruitful results from Minkowski's geometric viewpoint on number theory is his so called 1st Fundamental Theorem. It provides an optimal upper bound for the volume of an o-symmetric convex body whose only interior lattice…