Ubiquitous systems and metric number theory
| dc.creator | Durand, Arnaud | |
| dc.date | 2007-09-22 | |
| dc.date.accessioned | 2026-07-07T08:31:42Z | |
| dc.date.available | 2026-07-07T08:31:42Z | |
| dc.description | We investigate the size and large intersection properties of $$E_{t}=\{x\in\R^d \:|\: \|x-k-x_{i}\|<{r_{i}}^t\text{for infinitely many}(i,k)\in I^{μ,α}\times\Z^d\},$$ 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 $I^{μ,α}$ denotes the set of all $i\in I$ such that the $μ$-mass of the ball with center $x_{i}$ and radius $r_{i}$ behaves as ${r_{i}}^α$ for a given Borel measure $μ$ and a given $α>0$. 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 $μ$. 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. | |
| dc.description | 23 pages | |
| dc.identifier | https://arxiv.org/abs/0709.3595 | |
| dc.identifier | http://arxiv.org/abs/0709.3595 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/138577 | |
| dc.subject | Metric Geometry | |
| dc.subject | Number Theory | |
| dc.subject | 28A80; 28A78; 11J83 | |
| dc.title | Ubiquitous systems and metric number theory | |
| dc.type | text |