  Our main result marks progress on an old conjecture of Vitushkin. We show
that a compact set in the plane with plenty of big projections (PBP) has
positive analytic capacity, along with a quantitative lower bound. A higher
dimensional counterpart is also proved for capacities related to the Riesz
kernel, including the Lipschitz harmonic capacity. The proof uses a
construction of a doubling Frostman measure on a lower content regular set,
which may be of independent interest. Our second main result is the Analyst's
Traveling Salesman Theorem for sets with plenty of big projections. As a
corollary, we obtain a lower bound for the Hausdorff dimension of uniformly
wiggly sets with PBP. The second corollary is an estimate for the capacities of
subsets of sets with PBP, in the spirit of the quantitative solution to
Denjoy's conjecture.
