Related papers: Zerofree region for exponenetial sums

The general expression of the angular distance between two point sources as measured by an arbitrary observer is given. The modelling presented here is rigorous, covariant and valid in any space-time. The sources of light may be located at…

We prove that the upper bound for the van der Corput property of the set of perfect squares is O((log n)^{-1/3}), giving an answer to a problem considered by Ruzsa and Montgomery. We do it by constructing non-negative valued, normed…

We prove a theorem about approximation to an irrational number by rational numbers whose denominator n is free of prime factors bigger than a power of log n. We strengthen the result in version 1 by using an exponential sum over smooth…

Let A be a finite set of integers and F_A its exponential sum. McGehee, Pigno & Smith and Konyagin have independently proved that the L^1-norm of F_A is at least c log|A| for some absolute constant c. The lower bound has the correct order…

We show that a set of non-negative reals is the distance set of a separable complete metric space if and only if it is either countable or is an analytic set which has 0 as a limit point. We also consider spaces with simpler distance sets.

In a convex n-gon, let d[1] > d[2] > ... denote the set of all distances between pairs of vertices, and let m[i] be the number of pairs of vertices at distance d[i] from one another. Erdos, Lovasz, and Vesztergombi conjectured that m[1] +…

A detailed, internal symmetry exists between individual terms $n^{-s}$, where $n \in P$ is less than a particular value $n_p$, and sums over conjugate regions consisting of adjoining steps $n$ greater than $n_p$. The boundaries of the…

We consider a translation and dilation invariant system consisting of k diagonal equations of degrees 1,2,...,k with integer coefficients in s variables, where s is sufficiently large in terms of k. We show via the Hardy-Littlewood circle…

We give an example of an exact, stably finite, simple. separable C*-algebra D which is not isomorphic to its opposite algebra. Moreover, D has the following additional properties. It is stably finite, approximately divisible, has real rank…

Finding an exact maximum distance of two points in the given set is a fundamental computational problem which is solved in many applications. This paper presents a fast, simple to implement and robust algorithm for finding this maximum…

We study exponential sums of the form $\sum_{n=1}^N e^{2\pi i a b^n/m}$ for non-zero integers $a,b,m$. Classically, non-trivial bounds were known for $N\ge \sqrt{m}$ by Korobov, and this range has been extended significantly by Bourgain as…

For a finite abelian group $G$ and a positive integer $k$, let $s_{k}(G)$ denote the smallest integer $\ell\in\mathbb{N}$ such that any sequence $S$ of elements of $G$ of length $|S|\geq\ell$ has a zero-sum subsequence with length $k$. The…

Let $C_n$ be the cyclic group of order $n$. In this paper, we provide the exact values of some zero-sum constants over $C_n \rtimes_s C_2$ where $s \not\equiv \pm1 \pmod n$, namely $\eta$-constant, Gao constant, and Erd\H{o}s-Ginzburg-Ziv…

Let $\mathscr{H}$ be a family of digraphs. A digraph $D$ is \emph{$\mathscr{H}$-free} if it contains no isomorphic copy of any member of $\mathscr{H}$. For $k\geq2$, we set $C_{\leq k}=\{C_{2}, C_{3},\ldots,C_{k}\}$, where $C_{\ell}$ is a…

We are interested in finding an explicit estimate to the binomial sum $Q_n(x)=\sum_{k=0}^{n} k! {n\choose k}^2 (-x)^{k}$ at $x=1$ for $n=0,1,2,\ldots$. Despite of its own interest the polynomial $Q_n(x)$ is important as the denominator in…

We consider the number of distinct distances between two finite sets of points in ${\bf R}^k$, for any constant dimension $k\ge 2$, where one set $P_1$ consists of $n$ points on a line $l$, and the other set $P_2$ consists of $m$ arbitrary…

Let $A \in \mathbb{R}^{n \times (n - d)}$ be a random matrix with independent uniformly anti-concentrated entries satisfying $\mathbb{E}\lvert A\rvert_{HS}^2 \leq Kn(n-d)$ and let $H$ be the subspace spanned by the columns of $A$. Let $X…

If $\a$ is an irrational number, we let $\{p_n/q_n\}_{n\geq 0}$, be the approximants given by its continued fraction expansion. The Bruno series $B(\a)$ is defined as $$B(\a)=\sum_{n\geq 0} \frac{\log q_{n+1}}{q_n}.$$ The quadratic…

Let $a$ and $b$ be two distinct fixed positive integers such that $\min \{a,b\}>1.$ First, we correct an oversight from \cite{X-Z}. Then, we show that the equation in the title with $b \equiv 3 \pmod 8$, $b$ prime and $a$ even has no…

The Erd\H os unit distance conjecture in the plane says that the number of pairs of points from a point set of size $n$ separated by a fixed (Euclidean) distance is $\leq C_{\epsilon} n^{1+\epsilon}$ for any $\epsilon>0$. The best known…