Dimension of Union of Polygons

Description

By a classical and deep result of Bourgain and Marstrand, if a subset B of the plane contains a circle with center in every point of the plane then the set B must have positive Lebesgue measure. In the paper "Squares and their centers" https://arxiv.org/abs/1408.1029 we studied what happens if circles are replaced by squares with axes-parallel sides. It turned out that in this case the situation is completely different: a set B which contains an axis-parallel square with center in every point of the plane can be very small: it can have Lebesgue measure zero, even its fractal dimension can be smaller than 2. Here fractal dimension can be Hausdorff dimension, box counting dimension and packing dimension. (You don't have to know these notions in advance.) It turned out that the minimal Hausdorff dimension is 1 but the minimal box counting and packing dimension is 7/4.

The goal of this project is to generalize some of the results of this paper to other polygons. We will study only the box counting and packing dimension versions because it is known that the Hausdorff dimension of such a set can be 1 for any polygon. For box counting and packing dimension only very few partial results are known, and it does not seem to be hopeless at all to find more.

To estimate dimensions, especially box counting and somewhat also packing dimension, one often has to consider some discrete variants. Because of this in this project we will need to mix ideas form different areas like analysis, combinatorics and geometry and sometimes even number theory or probability theory might be useful.

The attached preliminary assignment is a warm up: you can learn a bit about the fractal dimensions we need and you might try to prove a known special case of the problem.

A few years ago in a similar BSM researh project, a former BSM student successfully generalized our results about squeres to cubes of higher dimenison, see
Riley Thorton Cubes and their centers, Acta Mathematica Hungarica, August 2017, Volume 152, Issue 2, pp 291–313; (arxiv version)

Two more papers that were written in similar BSM research projects under my supervision:

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