Related papers: Bounds on the Minkowski constants and a function i…

If $G$ is a finite Abelian group, define $s_{k}(G)$ to be the minimal $m$ such that a sequence of $m$ elements in $G$ always contains a $k$-element subsequence which sums to zero. Recently Bitz et al. proved that if $n = exp(G)$, then…

Two results concerning the number of threshold functions $P(2, n)$ and the probability ${\mathbb P}_n$ that a random $n\times n$ Bernoulli matrix is singular are established. We introduce a supermodular function $\eta^{\bigstar}_n : 2^{{\bf…

A Hankel operator $\mathbf{H}_\varphi$ on the Hardy space $H^2$ of the unit circle with analytic symbol $\varphi$ has minimal norm if $\|\mathbf{H}_\varphi\|=\|\varphi \|_2$ and maximal norm if $\|\mathbf{H}_\varphi\| = \|\varphi\|_\infty$.…

A Boolean function $f$ on $n$ variables is said to be a bent function if the absolute value of all its Walsh coefficients is $2^{n/2}$. Our main result is a new asymptotic lower bound on the number of Boolean bent functions. It is based on…

Using graph-theoretic methods we give a new proof that for all sufficiently large $n$, there exist sphere packings in $\R^n$ of density at least $cn2^{-n}$, exceeding the classical Minkowski bound by a factor linear in $n$. This matches up…

We study the problem of the existence of increasing and continuous solutions $\varphi\colon[0,1]\to[0,1]$ such that $\varphi(0)=0$ and $\varphi(1)=1$ of the functional equation \begin{equation*}…

We give several new applications of our theorem on the existence of multiplicity of graded families of ideals as a limit, including a very general Minkowski type inequality for graded families of ideals, a very general formula for existence…

It has been long--conjectured that the crossing number cr(K_m,n) of the complete bipartite graph K_m,n equals the Zarankiewicz Number Z(m,n):= floor((m-1)/2) floor(m/2) floor((n-1)/2) floor(n/2). Another long--standing conjecture states…

We show that common choices of kernel functions for a highly accurate and massively scalable nearest-neighbour based GP regression model (GPnn: \cite{GPnn}) exhibit gradual convergence to asymptotic behaviour as dataset-size $n$ increases.…

Wilf partitions are partitions of an integer $n$ in which all nonzero multiplicities are distinct. On his webpage, the late Herbert Wilf posed the problem to find "any interesting theorems" about the number f(n) of those partitions.…

We define the $k$-dimensional generalized Euler function $\varphi_k(n)$ as the number of ordered $k$-tuples $(a_1,\ldots,a_k)\in {\Bbb N}^k$ such that $1\le a_1,\ldots,a_k\le n$ and both the product $a_1\cdots a_k$ and the sum $a_1+\cdots…

In this paper, we consider the relationship between the Mahler measure of a polynomial and its separation. In 1964, Mahler proved that if $f(x) \in \mathbb{Z}[x]$ is separable of degree $n$, then $\operatorname{sep}(f) \gg_n M(f)^{-(n-1)}$.…

Minkowski's question mark function is the distribution function of a singular continuous measure: we study this measure from the point of view of logarithmic potential theory and orthogonal polynomials. We conjecture that it is regular, in…

For positive integers $m$ and $n$, the Zarankiewicz number $Z_{2,2}(m,n)$ can be defined as the maximum total degree of a linear hypergraph with $m$ vertices and $n$ edges. Guy determined $Z_{2,2}(m,n)$ for all $n \geq \binom{m}{2}/3+O(m)$.…

For a finite abelian group $G$, the Erd\H{o}s-Ginzburg-Ziv constant $\mathfrak{s}(G)$ is the smallest $s$ such that every sequence of $s$ (not necessarily distinct) elements of $G$ has a zero-sum subsequence of length…

Minkowski's 2nd theorem in the Geometry of Numbers provides optimal upper and lower bounds for the volume of a $o$-symmetric convex body in terms of its successive minima. In this paper we study extensions of this theorem from two different…

For every integer $n \geq 3$, we prove that the n-sublinear generalization of the Bi-Carleson operator of Muscalu, Tao, and Thiele given by nC^{\vec{\alpha}} :(f_1,..., f_n) \mapsto \sup_{M} \left| \int_{\vec{\xi} \cdot \vec{\alpha} >0,…

We use a characterization of Minkowski measurability to study the asymptotics of best packing on cut-out subsets of the real line with Minkowski dimension $d\in(0,1)$. Our main result is a proof that Minkowski measurability is a sufficient…

The thesis gave a fine study on the distribution of the coefficients of automorphic L-functions for GL(m) with m>1. In particular we have treated two types of problems: change of signs of these coefficients (when they are real) and their…

In this paper, we study the reducing subspaces for the multiplication operator by a finite Blaschke product $\phi$ on the Dirichlet space $D$. We prove that any two distinct nontrivial minimal reducing subspaces of $M_\phi$ are orthogonal.…