Hausdorff dimension of restricted Kakeya sets · arXivDesk2505.05709May 9, 2025minor updates and improvements
Hausdorff dimension of restricted Kakeya sets
Jonathan M. Fraser, Lijian Yang
Abstract
A Kakeya set in Rn is a compact set that contains a unit line segment Ie in each direction e∈Sn−1
. The Kakeya conjecture states that any Kakeya set in
has Hausdorff dimension
. We consider a restricted case where the midpoint of each line segment
must belong to a fixed set
with packing dimension at most
s∈[0,n] . In this case, we show that the Hausdorff dimension of the Kakeya set is at least
. Furthermore, using the "bush argument", we improve the lower bound to
max{n−s,n−gn(s)} , where
is defined inductively. For example, when
, we prove that the Hausdorff dimension is at least
max{519−53s,4−s} . We also establish Kakeya maximal function analogues of these results.