Related papers: Simultaneous approximation to a real number and to…

In 1974, M. B. Nathanson proved that every irrational number $\alpha$ represented by a simple continued fraction with infinitely many elements greater than or equal to $k$ is approximable by an infinite number of rational numbers $p/q$…

Let $q\ge 3$ be a non-exceptional modulus $q\ge3$, and let $a$ be a positive integer coprime with $q$. For any $\epsilon>0$, there exists $\alpha>0$ (computable), such that for all $x\ge \alpha (\log q)^2$, the interval $\left[…

We survey key techniques and results from approximation theory in the context of uniform approximations to real functions such as e^{-x}, 1/x, and x^k. We then present a selection of results demonstrating how such approximations can be used…

Assuming an array that consists of two parallel lines of qubits and that permits only nearest neighbor interactions, we construct physical and logical circuitry to enable universal fault tolerant quantum computation under the [[7,1,3]]…

It is shown in this note that approximating the number of independent sets in a $k$-uniform linear hypergraph with maximum degree at most $\Delta$ is NP-hard if $\Delta\geq 5\cdot 2^{k-1}+1$. This confirms that for the relevant sampling and…

Given a rational number $x$ and a bound $\varepsilon$, we exhibit $m,n$ such that $|x-12 s(m,n)|<\varepsilon$. Here $s(m,n)$ is the classical Dedekind sum and the parameters $m$ and $n$ are completely explicit in terms of $x$ and…

Let $\mathbb{F}_q$ denote the finite field with $q$ elements. In this work, we use characters to give the number of rational points on suitable curves of low degree over $\mathbb{F}_q$ in terms of the number of rational points on elliptic…

In this paper, we consider a problem of counting rational points near self-similar sets. Let $n\geq 1$ be an integer. We shall show that for some self-similar measures on $\mathbb{R}^n$, the set of rational points $\mathbb{Q}^n$ is…

Let $q$ be a perfect power of a prime number $p$ and $E({\mathbb F}_q)$ be an elliptic curve over ${\mathbb F}_q$ given by the equation $y^2=x^3+Ax+B$. For a positive integer $n$ we denote by $ \# E({\mathbb F}_{q^n})$ the number of…

We give an explicit formula for the expectation of the number of real lines on a random invariant cubic surface, i.e. a surface $Z\subset \mathbb{R}P^3$ defined by a random gaussian polynomial whose probability distribution is invariant…

Given a large finite point set, $P\subset \mathbb R^2$, we obtain upper bounds on the number of triples of points that determine a given pair of dot products. That is, for any pair of positive real numbers, $(\alpha, \beta)$, we bound the…

We derive exact and sharp lower bounds for the number of monochromatic generalized Schur triples $(x,y,x+ay)$ whose entries are from the set $\{1,\dots,n\}$, subject to a coloring with two different colors. Previously, only asymptotic…

Diophantine approximation is the problem of approximating a real number by rational numbers. We propose a version of this in which the numerators are approximately related to the denominators by a Laurent polynomial. Our definition is…

We investigate the uniform approximation provided by least squares polynomials on the unit Euclidean sphere $\mathbb{S}^q$ in $\mathbb{R}^{q+1}$, with $q\ge 2$. Like any other polynomial projection, the study concerns the growth, as the…

We investigate the asymptotic number of equivalence classes of linear codes with prescribed length and dimension. While the total number of inequivalent codes of a given length has been studied previously, the case where the dimension…

The papers shows an algorithm to search for approximations of reals to rationals of the form a/b^2 that runs on \sqrt(b) polynomial time steps.

We present lower estimates for the best constant appearing in the weak $(1,1)$ maximal inequality in the space $(\R^n,\|\cdot\|_{\iy})$. We show that this constant grows to infinity faster than $(\log n)^{1-o(1)}$ when $n$ tends to…

The $(u,v)$-Pad\'e approximation to a function $f$ is the (unique, up to scaling) rational approximation $f(x) = P(x)/Q(x) + O(x^{u+v+1})$, where $P$ has degree $u$ and $Q$ has degree $v$. Motivated by recent work of Molin, Pazuki, and…

It is well-known that the second-order cone can be outer-approximated to an arbitrary accuracy $\epsilon$ by a polyhedral cone of compact size defined by irrational data. In this paper, we propose two rational polyhedral…

In 1946 Erd\H os asked for the maximum number of unit distances, $u(n)$, among $n$ points in the plane. He showed that $u(n)> n^{1+c/\log\log n}$ and conjectured that this was the true magnitude. The best known upper bound is…