Related papers: The Duffin-Schaeffer Conjecture with extra diverge…

We show that for any $\epsilon<1$ and any $\mathcal{T}$ `drifting away from walls', Dirichlet's Theorem cannot be $\epsilon$-improved along $\mathcal{T}$ for Lebesgue almost every system of linear forms $Y$ (see the paper for definitions).…

In this article, we present a concavity property of the minimal $L^{2}$ integrals related to multiplier ideal sheaves with Lebesgue measurable gain on weakly pseudoconvex K\"ahler manifolds. As applications, we give a necessary condition…

Phenomena with a constrained sample space appear frequently in practice. This is the case e.g. with strictly positive data and with compositional data, like percentages and the like. If the natural measure of difference is not the absolute…

New cases of the multiplicity conjecture are considered.

We prove the Invariant Subspace Conjecture for separable Hilbert spaces.

Given quantities $\Delta_1,\Delta_2,\dots\geqslant 0$, a fundamental problem in Diophantine approximation is to understand which irrational numbers $x$ have infinitely many reduced rational approximations $a/q$ such that $|x-a/q|<\Delta_q$.…

Let $(\Omega, \mathcal{A}, \mu)$ be a probability space. The classical Borel-Cantelli Lemma states that for any sequence of $\mu$-measurable sets $E_i$ ($i=1,2,3,\dots$), if the sum of their measures converges then the corresponding…

Let $S$ be the set of subsequences $(x_{n_k})$ of a given real sequence $(x_n)$ which preserve the set of statistical cluster points. It has been recently shown that $S$ is a set of full (Lebesgue) measure. Here, on the other hand, we prove…

In order to give a unified generalization of the BW inequality and the DDVV inequality, Lu and Wenzel proposed three Conjectures 1, 2, 3 and an open Question 1 in 2016. In this paper we discuss further these conjectures and put forward…

A short and elementary proof of the joint convexity of relative entropy is presented, using nothing beyond linear algebra. The key ingredients are an easily verified integral representation and the strategy used to prove the Cauchy-Schwarz…

We present a proof of a multidimensional version of Peres-Schlag's theorem on Diophantine approximations with lacunary sequences.

We prove a conjecture of Kottwitz and Rapoport which implies a converse to Mazur's Inequality for all split and quasi-split (connected) reductive groups. These results are related to the non-emptiness of certain affine Deligne-Lusztig…

Building on work of Davenport and Schmidt, we mainly prove two results. The first one is a version of Gel'fond's transcendence criterion which provides a sufficient condition for a complex or $p$-adic number $\xi$ to be algebraic in terms…

In this paper we develop the convergence theory of simultaneous, inhomogeneous Diophantine approximation on manifolds. A consequence of our main result is that if the manifold $M \subset \mathbb{R}^n$ is of dimension strictly greater than…

In this two-part paper we prove an existence result for affine buildings arising from exceptional algebraic reductive groups. Combined with earlier results on classical groups, this gives a complete and positive answer to the conjecture…

We propose a new discretization of the mirror-Langevin diffusion and give a crisp proof of its convergence. Our analysis uses relative convexity/smoothness and self-concordance, ideas which originated in convex optimization, together with a…

A short, fairly self-contained proof is given of the Poincar\'e Conjecture. In the previous version there was an error on Page 8. This gap has now been filled.

In this note we prove a more general (and topological) version of Gr\"unbaum's conjecture about affine invariant points. As an application of our result we show that, if we consider the action of the group of similarities, Gr\"unbaum's…

In two dimensions, Gallagher's theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. We prove an inhomogeneous fibre version of Gallagher's theorem, sharpening and making unconditional a…

We provide an algorithm to approximate a finitely supported discrete measure $\mu$ by a measure $\nu_{N}$ corresponding to a set of $N$ points so that the total variation between $\mu$ and $\nu_N$ has an upper bound. As a consequence if…