Related papers: A slight improvement to Korenblum's constant

We show the existence of a constant $c > 0$ such that, for all positive integers $n$, there exist integers $1 \leq a_1 < \ldots < a_k \leq n$ such that there are at least $cn^2$ distinct integers of the form $\sum_{i=u}^{v}a_i$ with $1 \leq…

The Bohnenblust-Hille inequality was obtained in 1931 and (in the case of real scalars) asserts that for every positive integer $N$ and every $m$-linear mapping $T:\ell_{\infty}^{N}\times...\times\ell_{\infty}^{N}\rightarrow \mathbb{R}$ one…

We show that there is an absolute constant $c>0$ such that $|A+\lambda\cdot A|\geq e^{c\sqrt{\log |A|}}|A|$ for any finite subset $A$ of $\mathbb{R}$ and any transcendental number $\lambda\in\mathbb{R}$. By a construction of Konyagin and…

It is shown that the formula for the M\"obius pseudodistance for the annulus yields better estimates than previously known for the constant in the Bergman space maximum principle. A maximum principle for the Fock space is proved.

Let $f(z)=\sum_{n=1}^{\infty}a(n) e^{2\pi i nz}$ be a normalized Hecke eigenform in $S_{2k}^{\text{new}}(\Gamma_0(N))$ with integer Fourier coefficients. We prove that there exists a constant $C(f)>0$ such that any integer is a sum of at…

We prove that there is an absolute constant $c > 0$ such that every polynomial $P$ of the form $$P(z) = \sum_{j=0}^{n}{a_jz^j}\,, \quad |a_0| = 1\,, \quad |a_j| \leq M\,, \quad a_j \in \Bbb{C}\,, \quad M \geq 1\,,$$ has at most…

Kalton and Roberts [Trans. Amer. Math. Soc., 278 (1983), 803--816] proved that there exists a universal constant $K\leqslant 44.5$ such that for every set algebra $\mathcal{F}$ and every 1-additive function $f\colon \mathcal{F}\to \mathbb…

Given a $d$-tuple $T$ of commuting contractions on Hilbert space and a polynomial $p$ in $d$-variables, we seek upper bounds for the norm of the operator $p(T)$. Results of von Neumann and And\^o show that if $d=1$ or $d=2$, the upper bound…

An explicit upper bound is established for the least non-trivial integer zero of an arbitrary cubic form $C \in \mathbb{Z}[X_1,...,X_n],$ provided that $n \geq 14.$

We show that for any finite set $P$ of points in the plane and $\epsilon>0$ there exist $\displaystyle O\left(\frac{1}{\epsilon^{3/2+\gamma}}\right)$ points in ${\mathbb{R}}^2$, for arbitrary small $\gamma>0$, that pierce every convex set…

Various upper and lower bounds are provided for the (angular) Kronecker constants of sets of integers. Some examples are provided where the bounds are attained. It is proved that 5=16 bounds the angular Kronecker constants of 3-element sets…

Let $M_n$ be an $n\times n$ random matrix with i.i.d. Bernoulli(p) entries. We show that there is a universal constant $C\geq 1$ such that, whenever $p$ and $n$ satisfy $C\log n/n\leq p\leq C^{-1}$, \begin{align*} {\mathbb…

We prove that there exist infinitely many coprime numbers $a$, $b$, $c$ with $a+b=c$ and $c>\operatorname{rad}(abc)\exp(6.563\sqrt{\log c}/\log\log c)$. These are the most extremal examples currently known in the $abc$ conjecture, thereby…

It is known that there is a constant $c>0$ such that for every sequence $x_1, x_2,\ldots$ in $[0,1)$ we have for the star discrepancy $D^{*}_N$ of the first $N$ elements of the sequence that $N D^{*}_N\geq c\cdot \log N$ holds for…

Let $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$ be analytic in the unit disk with second coefficient $a_2$ satisfying $|a_2|=2b$, $0\leq b\leq1$. Sharp radius of Janowski starlikeness and other radius constants are obtained when $|a_n|\leq cn+d$…

In this paper we present a result about analytic functions f defined on the open unit disc and with a finite number of exceptional values containedin the real interval (0, 1). We find an upper bound for the modulus of f' in 0. This bound is…

Let u be a subharmonic function in D={|z|<1}. There exist an absolute constant C and an analytic function f in D such that \int_D |u(z)-log|f(z)|| dm(z)<C where m denotes the plane Lebesgue measure. We also consider uniform approximation.

The Heilbronn triangle problem asks for the placement of $n$ points in a unit square that maximizes the smallest area of a triangle formed by any three of those points. In $1972$, Schmidt considered a natural generalization of this problem.…

We estimate the number of zeros of a polynomial in $\mathbb{C}[z]$ within any small circular disc centered on the unit circle, which improves and comprehensively extends a result established by Borwein, Erd{\'e}lyi, and Littmann~\cite{BE1}…

For a subset A of a field F, write A(A + 1) for the set {a(b + 1):a,b\in A}. We establish new estimates on the size of A(A+1) in the case where F is either a finite field of prime order, or the real line. In the finite field case we show…