Related papers: Asymptotic complements in the integers

Symmetric elliptic integrals, which have been used as replacements for Legendre's integrals in recent integral tables and computer codes, are homogeneous functions of three or four variables. When some of the variables are much larger than…

We investigate asymptotic symmetries in flat backgrounds of dimension higher than or equal to four. For spin two we provide the counterpart of the extended BMS transformations found by Campiglia and Laddha in four-dimensional Minkowski…

Let $k \ge 2$ be an integer. We say a set $A$ of positive integers is an asymptotic basis of order $k$ if every large enough positive integer can be represented as the sum of $k$ terms from $A$. A set of positive integers $A$ is called…

We present two short proofs giving the best known asymptotic lower bound for the maximum element in a set of $n$ positive integers with distinct subset sums.

We use a Mellin-Barnes integral representation for the Lerch transcendent $\Phi(z,s,a)$ to obtain large $z$ asymptotic approximations. The simplest divergent asymptotic approximation terminates in the case that $s$ is an integer. For…

It is elementary and well-known that if an element x of a bounded modular lattice L has a complement in L then x has a relative complement in every interval [a,b] containing x. We show that the relatively strong assumption of modularity of…

Beta-integers (``$\beta$-integers'') are those numbers which are the counterparts of integers when real numbers are expressed in irrational basis $\beta > 1$. In quasicrystalline studies $\beta$-integers supersede the ``crystallographic''…

Non-compact proofs are a class of reasoning that is used in mathematics but overlooked in the analysis of (un)provability of consistency. We focus on proofs of arithmetical statements (*) "for any natural number n, F(n)." A proof of (*) is…

We prove that every subset of $\{1,\dots, N\}$ which does not contain any solutions to the equation $x+y+z=3w$ has at most $\exp(-c(\log N)^{1/5+o(1)})N$ elements, for some $c>0$. This theorem improves upon previous estimates. Additionally,…

The Weak Completion Semantics (WCS) is a computational cognitive theory that has shown to be successful in modeling episodes of human reasoning. As the WCS is a recently developed logic programming approach, this paper investigates the…

We give asymptotic estimates for the number of non-overlapping homothetic copies of some centrally symmetric oval $B$ which have a common point with a 2-dimensional domain $F$ having rectifiable boundary, extending previous work of the…

In connection with recent work on gaps in the asymptotic subranks of complex tensors the question arose whether the number of nonnegative real numbers that arise as the asymptotic subrank of some complex tensor is countable. In this short…

We show that there exists some $\delta > 0$ such that, for any set of integers $B$ with $B\cap[1,Y]\gg Y^{1-\delta}$ for all $Y \gg 1$, there are infinitely many primes of the form $a^2+b^2$ with $b\in B$. We prove a quasi-explicit formula…

We investigate properties of minimal $N$-point Riesz $s$-energy on fractal sets of non-integer dimension, as well as asymptotic behavior of $N$-point configurations that minimize this energy. For $s$ bigger than the dimension of the set…

We give more evidence for Patterson's conjecture on sums of exponential sums, by getting an asymptotic for a sum of quartic exponential sums over $\Q[i].$ Previously, the strongest evidence of Patterson's conjecture over a number field is…

Let $G$ be any group and $A$ be an arbitrary subset of $G$ (not necessarily symmetric and not necessarily containing the identity). The $h$-fold product set of $A$ is defined as $$A^{h} :=\lbrace a_{1}.a_{2}...a_{h} : a_{1},\ldots,a_n \in A…

In this paper, we obtain asymptotic formulae on nilmanifolds $\Gamma \backslash G$, wher $G$ is any stratified (or even graded) nilpotent Lie group equipped with a co-compact discrete subgroup $\Gamma$. We study especially the asymptotics…

We consider positive entropy $G$-systems for certain countable, discrete, infinite left-orderable amenable groups $G$. By undertaking local analysis, the existence of asymptotic pairs and chaotic sets will be studied in connecting with the…

Using ideas of S. Wassermann on non-exact $C^*$-algebras and property T groups, we show that one of his examples of non-invertible C*-extensions is not semi-invertible. To prove this, we show that a certain element vanishes in the…

Two observables are called complementary if preparing a physical object in an eigenstate of one of them yields a completely random result in a measurement of the other. We investigate small sets of complementary observables that cannot be…