This thesis deals with two topics: the Danzer problem in geometric discrepancy and the Sarkozy-Furstenberg theorem in combinatorics. The first part is devoted to the Danzer problem. A suitable weakening of its statement leads one to a problem of visibility in so-called dense forests. These are discrete point sets in the Euclidean space getting uniformly close to long enough line segments. This motivates the investigation of visibility concepts emerging from discrete geometry as well as the study of the distribution of sequences in the Euclidean space, the torus and the sphere. The following types of results are established: (1) the best known visibility bounds for dense forests are improved in any dimension, (2) geometrical and visibility concepts concerning planar spiral point sets are generalised to higher dimensions and (3) density properties of oscillating sequences in the real line are established. The second part concerns the Sarkozy-Furstenberg theorem. A multivariate version of the theorem is proved thanks to methods from Fourier analysis and with the help of a density increment argument.
|Date of Award||31 Dec 2022|
- The University of Manchester
|Supervisor||Faustin Adiceam (Supervisor), Gareth Jones (Supervisor) & Yuri Bazlov (Supervisor)|