Related papers: A quantitative version of Tao's result on the Toep…

We show that a square-tiling of a $p\times q$ rectangle, where $p$ and $q$ are relatively prime integers, has at least $\log_2p$ squares. If $q>p$ we construct a square-tiling with less than $q/p+C\log p$ squares of integer size, for some…

Let $k$ be an algebraic closure of a finite field of odd characteristic. We prove that for any rank two graded Higgs bundle with maximal Higgs field over a generic hyperbolic curve $X_1$ defined over $k$, there exists a lifting $X$ of the…

Let $L^p(\mathbf{T})$ be the Lesbegue space of complex-valued functions defined in the unit circle $\mathbf{T}=\{z: |z|=1\}\subseteq \mathbb{C}$. In this paper, we address the problem of finding the best constant in the inequality of the…

Answering a question of Benjamini & Schramm [8], we show that the Poisson boundary of any planar, uniquely absorbing (e.g. one-ended and transient) graph with bounded degrees can be realised geometrically as a circle, namely as the boundary…

In the 1970s, the collar theorem was proven, establishing the existence of uniform tubular neighborhoods of simple closed geodesics on compact surfaces, whose widths depend only on the lengths of the geodesics and the lower bound of the…

We consider a class $\mathscr{X}$ of continuous functions on $[0,1]$ that is of interest from two different perspectives. First, it is closely related to sets of functions that have been studied as generalizations of the Takagi function.…

In their solution to the orchard-planting problem, Green and Tao established a structure theorem which proves that in a line arrangement in the real projective plane with few double points, most lines are tangent to the dual curve of a…

We examine space-filling curves, which are surjective continuous maps from $[0,1]$ to some higher-dimensional space, usually the unit square $[0,1]^2$. In particular, we define Peano's curve and Lebesgue's curve, and state some of their…

We show that the largest size of a collection of simple closed curves pairwise intersecting at most once on an orientable surface of Euler characteristic $\chi$ grows quadratically in $|\chi|$. This resolves a longstanding question of…

The Loewner equation encrypts a growing simple curve in the plane into a real-valued driving function. We show that if the driving function $\lambda$ is in $C^{\beta}$ with $\beta>2$ (or real analytic) then the Loewner curve is in $C^{\beta…

Let $f$ be a holomorphic modular form of prime level $p$ and trivial nebentypus. We show that there exists a computable $\delta>0$, such that $$ L\left(\tfrac{1}{2},\mathrm{Sym}^2 f\right)\ll p^{\tfrac{1}{2}-\delta}, $$ with the implied…

A locally uniform random permutation is generated by sampling $n$ points independently from some absolutely continuous distribution $\rho$ on the plane and interpreting them as a permutation by the rule that $i$ maps to $j$ if the $i$th…

We obtain the Kato square root estimate for second order elliptic operators in divergence form with mixed boundary conditions on an open and possibly unbounded set in $\mathbb{R}^d$ under two simple geometric conditions: The Dirichlet…

The well-known problem stated by A. Meir and L. Moser consists in tiling the unit square with rectangles (details), whose side lengths equal $1/n\times 1/(n+1)$, where indices~$n$ range from 1 to infinity. Recently, Terence Tao has proved…

Let $X$ be a compact Riemann surface of genus $g$. Jacobi's inversion theorem states that the Abel-Jacobi map $\varphi : X^{(g)} \longrightarrow J(X)$ is surjective, where $X^{(g)}$ is the symmetric product of $X$ of degree $g$ and $J(X)$…

We propose (and prove under some restrictions) that the square class of the central value of the $L$-function of an everywhere unramified symplectic Galois representation is given by a universal cohomological formula. This phenomenon is…

Using our T1 theorem with an energy side condition allowing common point masses, we extend our previous work in arXiv:1310.4484v3 on one measure supported on a line, to include regular C(1,delta) curves and to permit common point masses. In…

We examine pairs of closed plane curves that have the same closing property as two conic sections in Poncelet's porism. We show how the vertex curve can be computed for a given envelope and vice versa. Our formulas are universal in the…

It is proved that for any mapping of a unit segment to a unit square, there is a pair of points of the segment for which the square of the Euclidean distance between their images exceeds the distance between them on the segment by at least…

We study surfaces in Euclidean space constructed by the sum of two curves or that are graphs of the product of two functions. We consider the problem to determine all these surfaces with constant Gauss curvature. We extend the results to…