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Register a new userErdH{o}s distinct distances in hyperbolic surfaces
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In this paper, we introduce the notion of geodesic cover for Fuchsian groups, which summons copies of fundamental polygons in the hyperbolic plane to cover pairs of representatives realizing distances in the corresponding hyperbolic surface. Then we use estimates of geodesic-covering numbers to study the distinct distances problem in hyperbolic surfaces. Especially, for $Y$ from a large class of hyperbolic surfaces, we establish the nearly optimal bound $geq c(Y)N/log N$ for distinct distances determined by any $N$ points in $Y$, where $c(Y)>0$ is some constant depending only on $Y$. In particular, for $Y$ being modular surface or standard regular of genus $ggeq 2$, we evaluate $c(Y)$ explicitly. We also derive new sum-product type estimates.
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For any cofinite Fuchsian group $Gammasubset {rm PSL}(2, mathbb{R})$, we show that any set of $N$ points on the hyperbolic surface $Gammabackslashmathbb{H}^2$ determines $geq C_{Gamma} frac{N}{log N}$ distinct distances for some constant $C_{Gamma}>0$ depending only on $Gamma$. In particular, for $Gamma$ being any finite index subgroup of ${rm PSL}(2, mathbb{Z})$ with $mu=[{rm PSL}(2, mathbb{Z}): Gamma ]<infty$, any set of $N$ points on $Gammabackslashmathbb{H}^2$ determines $geq Cfrac{N}{mulog N}$ distinct distances for some absolute constant $C>0$.
In this paper we obtain a new lower bound on the ErdH{o}s distinct distances problem in the plane over prime fields. More precisely, we show that for any set $Asubset mathbb{F}_p^2$ with $|A|le p^{7/6}$, the number of distinct distances determined by pairs of points in $A$ satisfies $$ |Delta(A)| gg |A|^{frac{1}{2}+frac{149}{4214}}.$$ Our result gives a new lower bound of $|Delta{(A)}|$ in the range $|A|le p^{1+frac{149}{4065}}$. The main tools we employ are the energy of a set on a paraboloid due to Rudnev and Shkredov, a point-line incidence bound given by Stevens and de Zeeuw, and a lower bound on the number of distinct distances between a line and a set in $mathbb{F}_p^2$. The latter is the new feature that allows us to improve the previous bound due Stevens and de Zeeuw.
Let $W_t$ denote the wheel on $t+1$ vertices. We prove that for every integer $t geq 3$ there is a constant $c=c(t)$ such that for every integer $kgeq 1$ and every graph $G$, either $G$ has $k$ vertex-disjoint subgraphs each containing $W_t$ as minor, or there is a subset $X$ of at most $c k log k$ vertices such that $G-X$ has no $W_t$ minor. This is best possible, up to the value of $c$. We conjecture that the result remains true more generally if we replace $W_t$ with any fixed planar graph $H$.
We develop a quantitative large deviations theory for random Bernoulli tensors. The large deviation principles rest on a decomposition theorem for arbitrary tensors outside a set of tiny measure, in terms of a novel family of norms generalizing the cut norm. Combined with associated counting lemmas, these yield sharp asymptotics for upper tails of homomorphism counts in the $r$-uniform ErdH{o}s--Renyi hypergraph for any fixed $rge 2$, generalizing and improving on previous results for the ErdH{o}s--Renyi graph ($r=2$). The theory is sufficiently quantitative to allow the density of the hypergraph to vanish at a polynomial rate, and additionally yields (joint) upper and lower tail asymptotics for other nonlinear functionals of interest.
In 1935, ErdH{o}s and Szekeres proved that $(m-1)(k-1)+1$ is the minimum number of points in the plane which definitely contain an increasing subset of $m$ points or a decreasing subset of $k$ points (as ordered by their $x$-coordinates). We consider their result from an on-line game perspective: Let points be determined one by one by player A first determining the $x$-coordinate and then player B determining the $y$-coordinate. What is the minimum number of points such that player A can force an increasing subset of $m$ points or a decreasing subset of $k$ points? We introduce this as the ErdH{o}s-Szekeres on-line number and denote it by $text{ESO}(m,k)$. We observe that $text{ESO}(m,k) < (m-1)(k-1)+1$ for $m,k ge 3$, provide a general lower bound for $text{ESO}(m,k)$, and determine $text{ESO}(m,3)$ up to an additive constant.
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