# Ioannis Tsokanos (Manchester) - Visibility properties in Spirals and Forests

Dates: | 8 February 2022 |
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Times: | 15:00 - 16:00 |

What is it: | Seminar |

Organiser: | Department of Mathematics |

Who is it for: | University staff, External researchers, Adults, Alumni, Current University students |

Speaker: | Ioannis Tsokanos |

Abstract : A spiral in the d-dimensional Euclidean space is a point set of the form S(U) = { k^{1/d+1} * u_k }_{k}, where U = (u_k)_{k} is a sequence in the (d-1)-dimensional unit sphere. A Peres-type forest F(a) is the planar point set containing all the points of the form (k , l + a_k) or (l + a_k , k) with k,l integers and where a = (a_k)_{k} is a sequence in the unit torus.

The talk is concerned with the study of the distributional properties of spirals and Peres-type forests in the Euclidean space. To characterize such distributional properties, we employ notions from discrete geometry and visibility concepts. For instance, a Delone set (in discrete geometry) is a point set which is simultaneously uniform discrete (there exists a positive infimum distance between any two points of the set) and relatively dense (the distance of the set from any point of the space is upper bounded). Also, a point set Y has an empty set of visible points if, for any point x in \R^{d} and any direction v in the (d-1)-dimensional sphere, the distance of Y from the half-line { x + tv}_{t > 0} is zero.

Both structures under consideration (spirals and Peres-type forests) are defined with the help of a sequence whose terms are chosen in a proper space (the sphere and the torus, respectively). This talk has two goals: (1) to provide necessary and sufficient conditions on a spherical (resp. on a toral) sequence for the spiral (resp. the Peres' forest) which it generates to satisfy the aforementioned visibility concepts and (2) to exploit number theoretical tools from the theories of distribution of sequences modulo one and Diophantine approximation to guarantee the existence of spiral structures (resp. Peres' forests) satisfying the properties of being Delone or having an empty set of visible points.

### Speaker

Ioannis Tsokanos

Organisation: University of Manchester