# A note on random coverings of tori

@article{Persson2013ANO, title={A note on random coverings of tori}, author={Tomas Persson}, journal={Bulletin of the London Mathematical Society}, year={2013}, volume={47} }

This note provides a generalization of a recent result by Järvenpää, Järvenpää, Koivusalo, Li, and Suomala (to appear) on the dimension of limsup‐sets of random coverings of tori. The result in this note is stronger in the sense that it provides also a large intersection property of the limsup‐sets, the assumptions are weaker, and it implies the result of Järvenpää, Järvenpää, Koivusalo, Li, and Suomala as a special case. The proof is based on a recent result by Persson and Reeve from 2013.

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## References

SHOWING 1-5 OF 5 REFERENCES

### A Frostman-Type Lemma for Sets with Large Intersections, and an Application to Diophantine Approximation

- MathematicsProceedings of the Edinburgh Mathematical Society
- 2014

Abstract We consider classes of subsets of [0, 1], originally introduced by Falconer, that are closed under countable intersections, and such that every set in the class has Hausdorff dimension at…

### Hausdorff dimension of affine random covering sets in torus

- MathematicsAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques
- 2014

We calculate the almost sure Hausdorff dimension of the random covering set lim supn→∞(gn + ξn) in d-dimensional torus T, where the sets gn ⊂ T are parallelepipeds, or more generally, linear images…

### On randomly placed arcs on the circle

- Mathematics
- 2010

We completely describe in terms of Hausdorff measures the size of the set of points of the circle that are covered infinitely often by a sequence of random arcs with given lengths. We also show that…