Related papers: Demailly's Conjecture for general and very general…

For any integers $d, n \geq 2$ and $1/({\min\{n,d\}})^{0.4999} < \varepsilon<1$, we show the existence of a set of $n$ vectors $X\subset \mathbb{R}^d$ such that any embedding $f:X\rightarrow \mathbb{R}^m$ satisfying $$ \forall x,y\in X,\…

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…

The Duffin-Schaeffer conjecture is a central open problem in metric number theory. Let $\psi~\mathbb{N} \mapsto \mathbb{R}$ be a non-negative function, and set $\mathcal{E}_n :=\bigcup \left( \frac{a - \psi(n)}{n},\frac{a+\psi(n)}{n}…

We prove the inhomogeneous generalization of the Duffin-Schaeffer conjecture in dimension $m \geq 3$. That is, given $\mathbf{y}\in \mathbb{R}^m$ and $\psi:\mathbb{N}\to\mathbb{R}_{\geq 0}$ such that $\sum (\varphi(q)\psi(q)/q)^m = \infty$,…

Consider a walk in the plane made of $n$ unit steps, with directions chosen independently and uniformly at random at each step. Rayleigh's theorem asserts that the probability for such a walk to end at a distance less than 1 from its…

We introduce the theory of div point sets, which aims to provide a framework to study the combinatoric nature of any set of points in general position on an Euclidean plane. We then show that proving the unsatisfiability of some first-order…

We extend Carpi's results by showing that Dejean's conjecture holds for n >= 30.

A celebrated conjecture due to De Giorgi states that any bounded solution of the equation $\Delta u + (1-u^2) u = 0 \hbox{in} \R^N $ with $\pp_{y_N}u >0$ must be such that its level sets $\{u=\la\}$ are all hyperplanes, {\em \bf at least}…

For any $i,j \ge 0$ with $i+j =1$, let $\bad(i,j)$ denote the set of points $(x,y) \in \R^2$ for which $ \max \{\|qx\|^{1/i}, \|qy\|^{1/j} \} > c/q $ for all $ q \in \N $. Here $c = c(x,y)$ is a positive constant. Our main result implies…

Generalizing the Zalcman conjecture given by $\vert a_n^2 - a_{2n-1}\vert \leq (n-1)^2$, Ma proposed and proved that the inequality $$\vert a_n a_m-a_{n+m-1}\vert \leq (n-1)(m-1), \quad m,n \in \mathbb{N},$$ holds for functions…

For all $k\geq 2$, we provide almost-sharp quantitative results for the $k$-dimensional Duffin-Schaeffer conjecture, analogous to recent developments in the 1-D case of Koukoulopoulos-Maynard-Yang. In particular, for…

Given a nonnegative function $\psi : \N \to \R $, let $W(\psi)$ denote the set of real numbers $x$ such that $|nx -a| < \psi(n) $ for infinitely many reduced rationals $a/n (n>0) $. A consequence of our main result is that $W(\psi)$ is of…

We establish the existence of $N$-point sets in dimension $d$ whose star-discrepancy is bounded above by $2.4631832 \sqrt{\frac{d}{N}}$, where the numerical constant improves upon all previously known bounds. This improvement is obtained by…

In this paper, we consider sums of generalized polygonal numbers with repeats, generalizing Fermat's polygonal number theorem which was proven by Cauchy. In particular, we obtain the minimal number of generalized $m$-gonal numbers required…

We prove the qualitative part of Demailly's conjecture on transcendental Morse inequalities for differences of two nef classes satisfying a numerical relative positivity condition on an arbitrary compact K\"ahler (and even more general)…

Refining an argument of the second author, we improve the known bounds for the number of rational points near a submanifold of $\mathbb{R}^d$ of intermediate dimension under a natural curvature condition. Furthermore, in the codimension $2$…

In a series of papers in the 1960's, S. G\"ahler defined and investigated so-called m-metric spaces and their topological properties. An m-metric assigns to any tuple of m+1 elements a real value (more generally an element in a partially…

Given a rational number $x$ and a bound $\varepsilon$, we exhibit $m,n$ such that $|x-12 s(m,n)|<\varepsilon$. Here $s(m,n)$ is the classical Dedekind sum and the parameters $m$ and $n$ are completely explicit in terms of $x$ and…

Let $V \subset \mathbb{R}$ be a finite set with $|V| = n $ and suppose we are given each pairwise distance independently with probability $p$. We show that if $p = (1+\epsilon)/n$, for some fixed $\epsilon >0$, then we can reconstruct a…

Fixed points for scalar theories in $4-\varepsilon$, $6-\varepsilon$ and $3-\varepsilon$ dimensions are discussed. It is shown how a large range of known fixed points for the four dimensional case can be obtained by using a general…