Related papers: Spiral Delone sets and three distance theorem

A systematic procedure is developed for constructing fermion systems in discrete space-time which have a given outer symmetry. The construction is illustrated by simple examples. For the symmetric group, we derive constraints for the number…

We prove results on the structure of a subset of the circle group having positive inner Haar measure and doubling constant close to the minimum. These results go toward a continuous analogue in the circle of Freiman's $3k-4$ theorem from…

We prove that a well-distributed subset of R^2 can have a separated distance set only if the distance is induced by a polygon.

The spontaneous breaking of an approximate discrete symmetry is considered, with the resulting protodomains of true and false vacuum being separated by domain walls. Given a strong, symmetric Yukawa coupling of the real scalar field to a…

We characterize equicontinuous Delone dynamical systems as those coming from Delone sets with strongly almost periodic Dirac combs. Within the class of systems with nite local complexity the only equicontinuous systems are then shown to be…

We establish an adelic version of Dirichlet's approximation theorem on spheres. Let $K$ be a number field, $E$ be a rigid adelic space over $K$ and $q\colon E\to K$ be a quadratic form. Let $v$ be a place of $K$ and $\alpha\in…

A fully implementable filtered polynomial approximation on spherical shells is considered. The method proposed is a quadrature-based version of a filtered polynomial approximation. The radial direction and the angular direction of the…

Let $F$ be Cayley's ruled cubic surface in a projective three-space over any commutative field $K$. We determine all collineations fixing $F$, as a set, and all cubic forms defining $F$. For both problems the cases $|K|=2,3$ turn out to be…

Fermat's principle and variational analysis is used to analyze the trajectories of light propagating in a radially inhomogeneous medium with a singularity in the center. It is found that the light trajectories are similar to those around a…

This paper reports on constructive approximation methods for three classes of holomorphic functions on the unit disk which are closely connected each other: the class of starlike and spirallike functions, the class of semigroup generators,…

We construct Salem sets in $\mathbb{R}/\mathbb{Z}$ of any dimension (including $1$) which do not contain any arithmetic progressions of length $3$. Moreover, the sets can be taken to be Ahlfors regular if the dimension is less than $1$, and…

For $ E\subset \mathbb{F}_q^d$, let $\Delta(E)$ denote the distance set determined by pairs of points in $E$. By using additive energies of sets on a paraboloid, Koh, Pham, Shen, and Vinh (2020) proved that if $E,F\subset \mathbb{F}_q^d $…

For a given real number $\alpha$, let us place the fractional parts of the points $0, \alpha, 2 \alpha,$ $ \cdots, (N-1) \alpha$ on the unit circle. These points partition the unit circle into intervals having at most three lengths, one…

We give a necessary and sufficient condition for a system of linear inhomogeneous fractional differential equations to have at least one bounded solution. We also obtain an explicit description for the set of all bounded (or decay)…

It can be shown in a solvable field theory model that the couplings of the composite vector bosons made of a fermion pair approach the gauge couplings in the limit of strong binding. Although this phenomenon may appear accidental and…

We simplify proof of the theorem that close to any pseudoholomorphic disk there passes a pseudoholomorphic disk of arbitrary close size with any pre-described sufficiently close direction. We apply these results to the Kobayashi and Hanh…

We show that if a finite point set $P\subseteq \mathbb{R}^2$ has the fewest congruence classes of triangles possible, up to a constant $M$, then at least one of the following holds. (1) There is a $\sigma>0$ and a line $l$ which contains…

Let $E\subset \mathbb R^n$, $n\ge 2$, be a set of finite perimeter with $|E|=|B|$, where $B$ denotes the unit ball. When $n=2$, since convexification decreases perimeter (in the class of open connected sets), it is easy to prove the…

We exploit dynamical properties of diagonal actions to derive results in Diophantine approximations. In particular, we prove that the continued fraction expansion of almost any point on the middle third Cantor set (with respect to the…

The Fermat-Steiner problem consists in finding all points in a metric space $Y$ such that the sum of distances from each of them to the points from some fixed finite subset of $Y$ is minimal. This problem is investigated for the metric…