An ordinary circle of a set P of n points in the plane is defined as a circle that contains exactly three points of P. We show that if P is not contained in a line or a circle, then P spans at least ordinary circles. Moreover, we determine the exact minimu ...
In the present thesis, we delve into different extremal and algebraic problems arising from combinatorial geometry. Specifically, we consider the following problems. For any integer n≥3, we define e(n) to be the minimum positive integer such that an ...
This thesis investigates the growth of the natural cocycle introduced by Klingler for the Steinberg representation. When possible, we extend the framework of simple algebraic groups over a local field to arbitrary Euclidean buildings. In rank one, the grow ...
Let P and Q be finite sets of points in the plane. In this note we consider the largest cardinality of a subset of the Minkowski sum S ⊆ P⊕Q which consist of convex independent points. We show that, if P and Q contain at most n points, then |S| = O(n^(4/3) ...
We show that the lines of every arrangement of n lines in the plane can be colored with O(root n/log n) colors such that no face of the arrangement is monochromatic. This improves a bound of Bose et al. by a circle minus(root/log n) factor. Any further imp ...
Erd\H{o}s conjectured in 1946 that every n-point set P in convex position in the plane contains a point that determines at least floor(n/2) distinct distances to the other points of P. The best known lower bound due to Dumitrescu (2006) is 13n/36 - O(1). I ...
In speckle interferometry (SI), temporal signals are amplitude- and frequency-modulated signals and exhibit a fluctuating background. Prior to phase computation, this background intensity must be eliminated. Here our approach is to build a complex signal f ...
