Related papers: Carleson's $\varepsilon^2$ conjecture in higher di…

We give a new proof of the well-known upper bound on the correlation function of a gas of non-overlapping dipoles of fixed length and discrete orientation working directly in the charge representation, instead of the more usual sine-Gordon…

In 1960, Gr\"{u}nbaum proved that for any convex body $C\subset\mathbb{R}^d$ and every halfspace $H$ containing the centroid of $C$, one has that the volume of $H\cap C$ is at least a $\frac{1}{e}$-fraction of the volume of $C$. Recently,…

Given a fractal $\mathcal{I}$ whose Hausdorff dimension matches with the upper-box dimension, we propose a new method which consists in selecting inside $\mathcal{I}$ some subsets (called quasi-Cantor sets) of almost same dimension and with…

We prove that if $A,B$ are compact subsets of $\mathbb{R}$ such that the upper density of $B$ is positive at every point of $B$, then there is a closed null set $N\subset A$ such that $N+B=A+B$. As a corollary we find that if $A,B\subset…

We investigate the small constant case of a characterization of $A_\infty$ weights due to Fefferman, Kenig and Pipher. In their work, Fefferman, Kenig and Pipher bound the logarithm of the $A_\infty$ constant by the Carleson norm of a…

J.F. Carlson conjectured in 1995 that if G is a finite group and k is a field whose characteristic p divides the order of G that the depth of H*(G,k) equals the minimum of the dimensions of associated primes of H*(G,k). This is obviously…

In this article, we present a simpler and alternative proof of the solvability of the regularity problem - that is, the Dirichlet problem with boundary data in $\dot W^{1,p}$ - for uniformly elliptic operators on $\mathbb{R}^n_+$ under a…

Carleson's theorem on the pointwise convergence of Fourier series provides bounds for a maximal operator, with the maximum taken over all choices of linear functions of a phase argument. We extend this to all quadratic choices of phase…

Suppose that $Z$ is a random closed subset of the hyperbolic plane $\H^2$, whose law is invariant under isometries of $\H^2$. We prove that if the probability that $Z$ contains a fixed ball of radius 1 is larger than some universal constant…

We prove a quantitative Borg-Levinson theorem for a large class of unbounded potentials. We give a detailed proof when the dimension of the space is greater than or equal to five. We also indicate the modifications necessary to cover lower…

We prove that the diagonal of the transition probabilities for the d-dimensional Bessel processes on (0, 1], reflected at 1, which we denote by $p_R^N(t, r,r)$, is an increasing function of r for d>2 and that this is false for d=2.

We show that the Hausdorff dimension of $(s,t)$-Furstenberg sets is at least $s+t/2+\epsilon$, where $\epsilon>0$ depends only on $s$ and $t$. This improves the previously best known bound for $2s<t\le 1+\epsilon(s,t)$, in particular…

A $k$-configuration is a collection of $k$ distinct integers $x_1,\ldots,x_k$ together with their pairwise arithmetic means $\frac{x_i+x_j}{2}$ for $1 \leq i < j \leq k$. Building on recent work of Filmus, Hatami, Hosseini and Kelman on…

We give conditions on a general family $P_{\lambda}:\R^n\to\R^m, \lambda \in \Lambda,$ of orthogonal projections which guarantee that the Hausdorff dimension formula $\dim A\cap P_{\lambda}^{-1}\{u\}=s-m$ holds generically for measurable…

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}…

Given a relatively compact set $\Omega \subseteq \mathbb{R}$ of Lebesgue measure $|\Omega|$ and $\varepsilon > 0$, we show the existence of a set $\Lambda \subseteq \mathbb{R}$ of uniform density $D (\Lambda) \leq (1+\varepsilon) |\Omega|$…

We prove that the Kakeya maximal conjecture is equivalent to the $\Omega$-Kakeya maximal conjecture. This completes a recent result in [2] where Keleti and Math{\'e} proved that the Kakeya conjecture is equivalent to the $\Omega$-Kakeya…

We investigate the relation between Carleson sequence and balayage, and use this to give an easy proof of the equivalence of the L1-norms of the maximal function and the square function in non-honogeneous martingale settings.

Let $z_1,\dots,z_n$ be complex numbers with $|z_j|\le \rho$, where $\rho>1$. Cassels proved that, under an additional restriction on $\rho$, the inequality \[ \prod_{j\ne k}\bigl|1-\overline{z_j}z_k\bigr| \le…

Let $M$ and $N$ be two closed hyperbolic Riemann surfaces. The Ehrenpreis Conjecture (proved by Kahn-Markovic) asserts that for any $\epsilon>0$ there are finite covers $M_\epsilon \to M$, and $N_\epsilon \to N$, such that the Teichmuller…