UBC Theses and Dissertations
Combinatorial problems for intervals, fractal sets and tubes Di Benedetto, Daniel
We discuss three problems related to combinatorial geometry, based on two separate works. The first work concerns arrangements of intervals in R² for which there are many pairs forming trapezoids, meaning the convex hull of the pair is a trapezoid. We characterise arrangements forming more than a certain number of trapezoids, showing that all such sets have underlying algebraic structure. An important role is played in particular by conic curves. The proof uses a transformation from intervals in the plane to lines in R³ and then relies on a theorem of Guth and Katz on intersecting lines in R³. The second work concerns combinatorial problems for discretised sets, where objects are only distinguishable up to some small scale. Discretised sets can be used to approximate fractal sets, and our results imply improved quantitative bounds for the 1/2-Furstenberg set problem in R² and the upper Minkowski dimension of Besicovitch sets in R³, as well as slight generalisations of each of these problems. The techniques involved in this second work are mostly combinatorial and our main ingredient is the discretised sum-product theorem from additive combinatorics. In particular, we reduce the 1/2-Furstenberg set problem to the discretised-sum product problem and reduce the Besicovitch set problem to the Furstenberg set problem.
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