In this thesis, we use the connections between projections and rectifiability to study problems in geometric measure theory, harmonic analysis, and complex analysis.
In \Cref{chapter:kakeya} (joint work with Marianna Cs\"ornyei), we study ``curved'' versions of the Kakeya needle problem and Besicovitch sets; i.e., we consider what happens if we replace the line segment with a $C^1$ curve, or more generally, a rectifiable set $E \subset \R^2$. Roughly speaking, our main result states that we can move a rectifiable set $E$ between any two prescribed positions in a set of measure zero, provided that at each time moment $t$ of the movement we are allowed to ``hide'' a set $E_t \subset E$ of linear measure zero. %solve the Kakeya needle problem and construct a Besicovitch and a Nikodym set for rectifiable sets.
In \Cref{chapter:skeleton} (joint work with Marianna Cs\"ornyei, Korn\'elia H\'era and Tam\'as Keleti), our aim is to find the minimal Hausdorff dimension of the union of scaled and/or rotated copies of the $k$-skeleton of a fixed polytope centered at the points of a given set. For many of these problems, we show that a typical arrangement in the sense of Baire category gives minimal Hausdorff dimension. In particular, this proves a conjecture of R.~Thornton. Our results also show that Nikodym sets are typical among all sets which contain, for every $x\in\R^n$, a punctured hyperplane $H\setminus \{x\}$ through $x$. With similar methods we also construct a Borel subset of $\Rn$
of Lebesgue measure zero containing a hyperplane at every positive distance from every point.
In \Cref{chapter:projections} (joint work with Xavier Tolsa), we study the connection between the analytic capacity of a set and the size of its orthogonal projections. More precisely, we prove that if $E\subset \C$ is compact and $\mu$ is a Borel measure supported on $E$, then the analytic capacity of $E$ satisfies
$$
\gamma(E) \geq c\,\frac{\mu(E)^2}{\int_I \|P_\theta\mu\|_2^2\,d\theta},
$$
where $c$ is some positive constant, $I\subset [0,\pi)$ is an arbitrary interval, and $P_\theta\mu$ is the image measure of $\mu$ by $P_\theta$, the orthogonal projection onto the line $\{re^{i\theta}:r\in\R\}$. This result is related to an old conjecture of Vitushkin about the relationship between the Favard length and analytic capacity.