Related papers: Sharp bounds for the Tao-Vu Discrete John's Theore…

We establish sharp bounds for the Hausdorff dimension of sets of irrational numbers in $(0,1)$ whose digits in the $N$-expansion are either uniformly bounded or tend to infinity. For sets with digits bounded by an integer $M \ge N$, we…

We derive a second-order ordinary differential equation (ODE) which is the limit of Nesterov's accelerated gradient method. This ODE exhibits approximate equivalence to Nesterov's scheme and thus can serve as a tool for analysis. We show…

In this project we show the existence of arbitrary length arithmetic progressions in model sets and Meyer sets in the Euclidean $d$-space. We prove a van der Waerden type theorem for Meyer sets. We show that pure point subsets of Meyer sets…

The main result of this paper is a proof of the continuity of a family of integral functionals defined on the space of functions of bounded variation with respect to a topology under which smooth functions are dense. These functionals occur…

Let X be a normed space. A subset A of X is approximately convex if $d(ta+(1-t)b,A) \le 1$ for all $a,b \in A$ and $t \in [0,1]$ where $d(x,A)$ is the distance of $x$ to $A$. Let $\Co(A)$ be the convex hull and $\diam(A)$ the diameter of…

We provide new upper and lower bounds on the minimum possible ratio of the spectral and Frobenius norms of a (partially) symmetric tensor. In the particular case of general tensors our result recovers a known upper bound. For symmetric…

Fix $d \ge 3$. We show the existence of a constant $c>0$ such that any graph of diameter at most $d$ has average distance at most $d-c \frac{d^{3/2}}{\sqrt n}$, where $n$ is the number of vertices. Moreover, we exhibit graphs certifying…

A new concept of a deformed numerical range $W^\rho(T)$ is introduced. Here $T$ is a bounded linear operator or a matrix and $ \rho \in[1,+\infty)$ is a parameter. Each $W^\rho(T)$ is a closed convex set that contains the spectrum of $T$.…

We prove an asymptotic formula for the shifted convolution of the divisor functions $d_k(n)$ and $d(n)$ with $k \geq 4$, which is uniform in the shift parameter and which has a power-saving error term, improving results obtained previously…

Let $G$ be a connected $n$-vertex graph in a proper minor-closed class $\mathcal G$. We prove that the extension complexity of the spanning tree polytope of $G$ is $O(n^{3/2})$. This improves on the $O(n^2)$ bounds following from the work…

Suppose $d+1$ absolutely continuous probability measures $m_0, \ldots, m_d$ on $\mathbb{R}^d$ are given. In this paper, we prove that there exists a point of $\mathbb{R}^d$ that belongs to the convex hull of $d+1$ points $v_0, \ldots, v_d$…

We show that once $\theta>17/30$, every sufficiently long interval $[x,x+x^\theta]$ contains many $k$-term arithmetic progressions of primes, uniformly in the starting point $x$. More precisely, for each fixed $k\ge3$ and $\theta>17/30$,…

We construct skew corner-free subsets of $[n]^2$ of size $n^2\exp(-O(\sqrt{\log n}))$, thereby improving on recent bounds of the form $\Omega(n^{5/4})$ obtained by Pohoata and Zakharov. In the other direction, we prove that any such set has…

We consider It\^o uniformly nondegenerate equations with time independent coefficients, the diffusion coefficient in $W^{1}_{d,loc}$, and the drift in $L_{d}$. We prove the unique strong solvability for any starting point and prove that as…

In 1984, Thomassen conjectured that for every constant $k \in \mathbb{N}$, there exists $d$ such that every graph with average degree at least $d$ contains a balanced subdivision of a complete graph on $k$ vertices, i.e. a subdivision in…

Tusn\'ady's problem asks to bound the discrepancy of points and axis-parallel boxes in $\mathbb{R}^d$. Algorithmic bounds on Tusn\'ady's problem use a canonical decomposition of Matou\v{s}ek for the system of points and axis-parallel boxes,…

We study the $L^p$ mapping properties of the strong spherical maximal function, which is a multiparameter generalisation of Stein's spherical maximal function. We show that this operator is bounded on $L^p$ for $p > 2$ in all dimensions $n…

In this paper, we prove the existence of fixed points of mappings satisfying the condition (Da), a kind of generalized nonexpansive mappings, on a weakly compact convex subset in a Banach space satisfying Opial's condition. And we use…

Let $A\subset [1, 2]$ be a $(\delta, \sigma)$-set with measure $|A|=\delta^{1-\sigma}$ in the sense of Katz and Tao. For $\sigma\in (1/2, 1)$ we show that $$ |A+A|+|AA|\gtrapprox \delta^{-c}|A|, $$ for…

The edge-of-the-wedge theorem in several complex variables gives the analytic continuation of functions defined on the poly upper half plane and the poly lower half plane, the set of points in $\mathbb{C}^d$ with all coordinates in the…