Related papers: Accumulation points of normalized approximations
We calculate normalization factors and reflection amplitudes in the W-invariant conformal quantum field theories. Using these CFT data we derive vacuum expectation values of exponential fields in affine Toda theories and related perturbed…
Let $S \subseteq \mathbb{N}$ have the property that for each $k \in S$ the set $(S - k) \cap \mathbb{N} \setminus S$ has asymptotic density $0$. We prove that there exists a basic sequence $Q$ where the set of numbers $Q$-normal of all…
Unified field theories act to merge the internal symmetries of the standard model into a single group. Here we lay out something different. That is, instead of aiming to unify the internal symmetries, we demonstrate a sense in which the…
Let $A$ be a set of natural numbers. A set $B$, a set of natural numbers, is said to be an additive complement of the set $A$ if all sufficiently large natural numbers can be represented in the form $x+y$, where $x\in A$ and $y\in B$. This…
We prove that coronizations with respect to arbitrary d-regular sets (not necessarily graphs) imply big pieces squared of these (approximating) sets. This is known (and due to David and Semmes in the case of sufficiently large co-dimension,…
Let $C$ be the attractor of the IFS $\{f_{d}(z) = (-n+i)^{-1}(z+d): d\in D\}$, $D\subset\{0, 1, \ldots, n^{2}\}$ and let $\dim$ denote the box-counting dimension. It is known that for all $\lambda\in[0, 1]$, that the set of complex numbers…
Point counting estimates are a key stepping stone to various results in metric Diophantine approximation. In this paper we use the quantitative non-divergence estimates originally developed by Kleinbock and Margulis to improve lower bounds…
We calculate the measure and Hausdorff dimension of sets of matrices over fields of formal power series with good approximation properties for a restricted set of denominators.
We prove that if a set is `large' in the sense of Erd\H{o}s, then it approximates arbitrarily long arithmetic progressions in a strong quantitative sense. More specifically, expressing the error in the approximation in terms of the gap…
In this paper, we study the Hausdorff dimension of self-similar measures and sets on the real line, where the generating iterated function system consists of some maps that share the same fixed point. In particular, we will show that out of…
This paper deals with two main topics related to Diophantine approximation. Firstly, we show that if a point on an algebraic variety is approximable by rational vectors to a sufficiently large degree, the approximating vectors must lie in…
We develop a quantitative theory of stochastic homogenization in the more general framework of differential forms. Inspired by recent progress in the uniformly elliptic setting, the analysis relies on the study of certain subadditive…
Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying Diophantine approximation in an arbitrary totally bounded metric space where the rationals are…
The type-PQ adjacency polytope associated to a simple graph is a $0/1$-polytope containing valuable information about an underlying power network. Chen and the first author have recently demonstrated that, when the underlying graph $G$ is…
The goal of this paper is to develop the theory of weighted Diophantine approximation of rational numbers to $p$-adic numbers. Firstly, we establish complete analogues of Khintchine's theorem, the Duffin-Schaeffer theorem and the…
We obtain that the nonzero accumulation points of the set of 3-fold canonical thresholds $\ct(X,S)$ are precisely $1/k$ where $k\ge 2$ is an integer and $S$ is an effective integral divisor of a projective 3-fold $X$ with only terminal…
We review our recent formulation of Colombeau type algebras as Hausdorff sequence spaces with ultranorms, defined by sequences of exponential weights. We extend previous results and give new perspectives related to echelon type spaces,…
The goal of this paper is to generalize the main results of [KM] and subsequent papers on metric Diophantine approximation with dependent quantities to the set-up of systems of linear forms. In particular, we establish `joint strong…
We show any subset $A\subset\mathbb{N}$ with positive upper Banach density contains the pattern $\{m,m+[n\alpha],\dots,m+k[n\alpha]\}$, for some $m\in\mathbb{N}$ and $n=p-1$ for some prime $p$, where…
We demonstrate $k+1$-term arithmetic progressions in certain subsets of the real line whose "higher-order Fourier dimension" is sufficiently close to 1. This Fourier dimension, introduced in previous work, is a higher-order (in the sense of…