Ubiquitous systems and metric number theory

Details Ubiquitous systems and metric number theory Ubiquitous systems and metric number theory

2007-09-22

arxiv.org/abs/0709.3595v1

Mathematics

Metric Geometry

23 pages

Scientific paper

We investigate the size and large intersection properties of $$E_{t}=\{x\in\R^d \:|\: \|x-k-x_{i}\|<{r_{i}}^t\text{for infinitely many}(i,k)\in I^{\mu,\alpha}\times\Z^d\},$$ where $d\in\N$, $t\geq 1$, $I$ is a denumerable set, $(x_{i},r_{i})_{i\in I}$ is a family in $[0,1]^d\times (0,\infty)$ and $I^{\mu,\alpha}$ denotes the set of all $i\in I$ such that the $\mu$-mass of the ball with center $x_{i}$ and radius $r_{i}$ behaves as ${r_{i}}^\alpha$ for a given Borel measure $\mu$ and a given $\alpha>0$. We establish that the set $E_{t}$ belongs to the class $\grint^h(\R^d)$ of sets with large intersection with respect to a certain gauge function $h$, provided that $(x_{i},r_{i})_{i\in I}$ is a heterogeneous ubiquitous system with respect to $\mu$. In particular, $E_{t}$ has infinite Hausdorff $g$-measure for every gauge function $g$ that increases faster than $h$ in a neighborhood of zero. We also give several applications to metric number theory.

Affiliated with

Also associated with

No associations

Rating

Ubiquitous systems and metric number theory does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Ubiquitous systems and metric number theory, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Ubiquitous systems and metric number theory will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-544287