  This thesis investigates two problems that are discrete analogues of two
harmonic analytic problems which lie in the heart of research in the field.
  More specifically, we consider discrete analogues of the maximal Kakeya
operator conjecture and of the recently solved endpoint multilinear Kakeya
problem, by effectively shrinking the tubes involved in these problems to
lines, thus giving rise to the problems of counting joints and multijoints with
multiplicities. In fact, we effectively show that, in $\mathbb{R}^3$, what we
expect to hold due to the maximal Kakeya operator conjecture, as well as what
we know in the continuous case due to the endpoint multilinear Kakeya theorem
by Guth, still hold in the discrete case.
  In particular, let $\mathfrak{L}$ be a collection of $L$ lines in
$\mathbb{R}^3$ and $J$ the set of joints formed by $\mathfrak{L}$, that is, the
set of points each of which lies in at least three non-coplanar lines of
$\mathfrak{L}$. It is known that $|J|=O(L^{3/2})$ (first proved by Guth and
Katz). For each joint $x\in J$, let the multiplicity $N(x)$ of $x$ be the
number of triples of non-coplanar lines through $x$. We prove here that
$$\sum_{x\in J} N(x)^{1/2}=O(L^{3/2}), $$while we also extend this result to
real algebraic curves in $\mathbb{R}^3$ of uniformly bounded degree, as well as
to curves in $\mathbb{R}^3$ parametrized by real univariate polynomials of
uniformly bounded degree.
  The multijoints problem is a variant of the joints problem, involving three
finite collections of lines in $\mathbb{R}^3$; a multijoint formed by them is a
point that lies in (at least) three non-coplanar lines, one from each
collection.
  We finally present some results regarding the joints problem in different
field settings and higher dimensions.
