Related papers: Asymptotically tight bounds on subset sums

We establish sharp estimates for the convergence rate of the Kranosel'ski\v{\i}-Mann fixed point iteration in general normed spaces, and we use them to show that the asymptotic regularity bound recently proved in [11] (Israel Journal of…

We give bounds on the error in the asymptotic approximation of the log-Gamma function $\ln\Gamma(z)$ for complex $z$ in the right half-plane. These improve on earlier bounds by Behnke and Sommer (1962), Spira (1971), and Hare (1997). We…

A long standing problem asks whether every group is sofic, i.e., can be separated by almost-homomorphisms to the symmetric group $Sym(n)$. Similar problems have been asked with respect to almost-homomorphisms to the unitary group $U(n)$,…

Let $G$ be a group and let $A\subseteq G$ be non-empty. We call $A$ an asymptotic $(r,l)$-approximate group if, for a fixed dilation factor $r$, the larger product sets $A^{hr}$ can, for all sufficiently large $h$, be covered by a bounded…

Schur introduced the problem on the smallest limit point for the arithmetic means of totally positive conjugate algebraic integers. This area was developed further by Siegel, Smyth and others. We consider several generalizations of the…

We show that a set $A \subset \{0,1\}^{n}$ with edge-boundary of size at most $|A| (\log_{2}(2^{n}/|A|) + \epsilon)$ can be made into a subcube by at most $(2 \epsilon/\log_{2}(1/\epsilon))|A|$ additions and deletions, provided $\epsilon$…

We investigate the spectral properties of the Dirichlet Laplacian on large finite metric balls within irregular infinite graphs of quadratic volume growth. We consider an exhaustion $G_n = B_{R_n}(x_0)$ and the spectral zeta value $Z_n(1) =…

We consider a one dimensional affine switched system obtained from a formal limit of a two dimensional linear system. We show this is equivalent to minimising the average digit in beta representations with unrestricted digits. We give a…

Let $r$ be a positive integer, and let $A$ be a nonempty finite set of at least two integers. We let $\tilde{C}_r(A)$ denote the {\em asymptotic $r$-covering number} of $A$, that is, the smallest integer value of $l$ for which, for all…

Let $G$ be a permutation group on a set $\Omega$. A subset of $\Omega$ is a base for $G$ if its pointwise stabiliser in $G$ is trivial. In this paper we introduce and study an associated graph $\Sigma(G)$, which we call the Saxl graph of…

We call a subset $A$ of the (additive) abelian group $G$ {\it $t$-independent} if for all non-negative integers $h$ and $k$ with $h+k \leq t$, the sum of $h$ (not necessarily distinct) elements of $A$ does not equal the sum of $k$ (not…

Let $\Omega \subset \mathbb{R}^N$, $N\ge 2$, be an open, connected, bounded set with $C^2$ boundary. In this paper we consider the torsion problem with Robin boundary conditions and we study the symmetry of the solutions when suitable extra…

In this paper, we establishe the extremal bounds of the topological indices -- Sigma index -- focusing on analyzing the sharp upper bounds and the lower bounds of the Sigma index, which is known $\sigma(G)=\sum_{uv\in…

In [dLMu05], DeLellis and M\"uller proved a quantitative version of Codazzi's theorem, namely for a smooth embedded surface $\ \Sigma \subseteq \mathbb{R}^3\ $ with area normalized to $\ {\cal H}^2(\Sigma) = 4 \pi\ $, it was shown that $\…

We show how to find upper and lower bounds to the median of a gamma distribution, over the entire range of shape parameter $k > 0$, that are the tightest possible bounds of the form $2^{-1/k} (A + Bk)$, with closed-form parameters $A$ and…

An asymptotic expansion for the generalised quadratic Gauss sum $$S_N(x,\theta)=\sum_{j=1}^{N} \exp (\pi ixj^2+2\pi ij\theta),$$ where $x$, $\theta$ are real and $N$ is a positive integer, is obtained as $x\rightarrow 0$ and…

We provide new upper and lower bounds on the minimum possible ratio of the spectral and Frobenius norms of a (partially) symmetric tensor. In the particular case of general tensors our result recovers a known upper bound. For symmetric…

Let $A=\{a_1,a_2,\dots, a_m\}$ be a subset of a finite abelian group $G$. We call $A$ {\it $t$-independent} in $G$, if whenever $$\lambda_1a_1+\lambda_2a_2+\cdots +\lambda_m a_m=0$$ for some integers $\lambda_1, \lambda_2, \dots ,…

This is a sequel to the paper arXiv:1312.6438 by the same authors. In this sequel, we quantitatively improve several of the main results of arXiv:1312.6438, and build on the methods therein. The main new results is that, for any finite set…

A finitely presented, torsion free, abelian-by-cyclic group can always be written as an ascending HNN extension Gamma_M of Z^n, determined by an n x n integer matrix M with det(M) \ne 0. The group Gamma_M is polycyclic if and only if…