ﻻ يوجد ملخص باللغة العربية
A circle, centered at the origin and with radius chosen so that it has non-empty intersection with the integer lattice $mathbb{Z}^{2}$, gives rise to a probability measure on the unit circle in a natural way. Such measures, and their weak limits, are said to be attainable from lattice points on circles. We investigate the set of attainable measures and show that it contains all extreme points, in the sense of convex geometry, of the set of all probability measures that are invariant under some natural symmetries. Further, the set of attainable measures is closed under convolution, yet there exist symmetric probability measures that are not attainable. To show this, we study the geometry of projections onto a finite number of Fourier coefficients and find that the set of attainable measures has many singularities with a fractal structure. This complicated structure in some sense arises from prime powers - singularities do not occur for circles of radius $sqrt{n}$ if $n$ is square free.
We consider the proportion of generalized visible lattice points in the plane visited by random walkers. Our work concerns the visible lattice points in random walks in three aspects: (1) generalized visibility along curves; (2) one random walker vis
Under the formalism of annealed averaging of the partition function, two types of random multifractal measures with their probability of multipliers satisfying power distribution and triangular distribution are investigated mathematically. In these t
This paper concerns the number of lattice points in the plane which are visible along certain curves to all elements in some set S of lattice points simultaneously. By proposing the concept of level of visibility, we are able to analyze more carefull
We call a polynomial monogenic if a root $theta$ has the property that $mathbb{Z}[theta]$ is the full ring of integers in $mathbb{Q}(theta)$. Consider the two families of trinomials $x^n + ax + b$ and $x^n + cx^{n-1} + d$. For any $n>2$, we show that
We show that in an L-annularly linearly connected, N-doubling, complete metric space, any n points lie on a K-quasi-circle, where K depends only on L, N and n. This implies, for example, that if G is a hyperbolic group that does not split over any vi