  We study a variety of problems about homothets of sets related to the Kakeya
conjecture. In particular, we show many of these problems are equivalent to the
arithmetic Kakeya conjecture of Katz and Tao. We also provide a proof that the
arithmetic Kakeya conjecture implies the Kakeya conjecture for packing
dimension, as this implication was previously only known for Minkowski
dimension.
  We consider several questions analogous to the classical results of Stein and
Bourgain about the Lebesgue measure of a set containing a sphere centered at
every point of $[0,1]^n$, where we replace spheres by arbitrary polytopes. We
give a lower bound for polytopes in $\mathbb{R}^n$, and show that this is sharp
for simplices. Finally, we generalize number theoretic methods of Green and
Ruzsa to study patterns in number fields and thereby provide upper bounds for
several of these homothet problems.
