Analytic capacity and dimension of sets with plenty of big projections

Abstract

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.

arXiv commentv2: major reorganization of the paper to make the proofs clearer, main results unchanged; Proposition 3.3 on coronization of lower content regular sets by Ahlfors regular sets is new (it was implicit in v1); to appear in Trans. Amer. Math. Soc.; 56 pages; supersedes arXiv:2109.10087

References in corpus (10)

• Existence of doubling measures via generalised nested cubes
• Characterising the big pieces of Lipschitz graphs property using projections
• Subsets of Rectifiable curves in Hilbert Space-The Analyst's TSP
• Plenty of big projections imply big pieces of Lipschitz graphs
• Hausdorff dimension of wiggly metric spaces
• Quantitative Comparisons of Multiscale Geometric Properties
• Higher dimensional Jordan curves
• Structure of sets with nearly maximal Favard length
• Metric properties of mean wiggly continua
• Birth and life of the $L^2$ boundedness of the Cauchy Integral on Lipschitz graphs

Cited by in corpus (1)

• Interactions between quantitative rectifiability, singular integrals, and boundary value problems for harmonic functions