- Published 2011

The thesis addresses problems from the field of geometric measure theory. It turns out<lb>that discrete methods can be used efficiently to solve these problems. Let us summarize<lb>the main results of the thesis.<lb>In Chapter 2 we investigate the following question proposed by Tamás Keleti. How<lb>large (in terms of Hausdorff dimension) can a compact set A ⊂ R be if it does not<lb>contain some given angle α, that is, it does not contain distinct points P,Q,R ∈ A with<lb>∠PQR = α? Or equivalently, how large dimension guarantees that our set must contain<lb>α?<lb>We also study an approximate version of this problem, where we only want our set to<lb>contain angles close to α rather than contain the exact angle α. This version turns out to<lb>be completely different from the original one, which is best illustrated by the case α = π/2.<lb>If the dimension of our set is greater than 1, then it must contain angles arbitrarily close<lb>to π/2. However, if we want to make sure that it contains the exact angle π/2, then we<lb>need to assume that its dimension is greater than n/2.<lb>Another interesting phenomenon is that different angles show different behaviour. In<lb>the approximate version the angles π/3, π/2 and 2π/3 play special roles, while in the<lb>original version π/2 seems to behave differently than other angles.<lb>The investigation of the above problems led us to the study of the so-called acute sets. A<lb>finite setH in R is called an acute set if any angle determined by three points ofH is acute.<lb>Chapter 3 of the thesis studies the maximal cardinality α(n) of an n-dimensional acute<lb>set. The exact value of α(n) is known only for n ≤ 3. For each n ≥ 4 we improve on the<lb>best known lower bound for α(n). We present different approaches. On one hand, we give<lb>a probabilistic proof that α(n) > c · 1.2. (This improves a random construction given by<lb>Erdős and Füredi.) On the other hand, we give an almost exponential constructive example<lb>which outdoes the random construction in low dimension (n ≤ 250). Both approaches<lb>use the small dimensional examples that we found partly by hand (n = 4, 5), partly by<lb>computer (6 ≤ n ≤ 10).<lb>Finally, in Chapter 4 we show that the Koch curve is tube-null, that is, it can be<lb>covered by strips of arbitrarily small total width. In fact, we prove the following stronger<lb>result: the Koch curve can be decomposed into three sets such that each can be projected<lb>to a line in such a way that the image has Hausdorff dimension less than 1. The proof<lb>contains geometric, combinatorial, algebraic and probabilistic arguments.

@inproceedings{Harangi2011DiscreteMI,
title={Discrete methods in geometric measure theory},
author={Viktor Harangi and Mikl{\'o}s Laczkovich and Andr{\'a}s Szűcs and Tam{\'a}s Keleti},
year={2011}
}