Decoupling and Near-optimal Restriction Estimates for Cantor Sets

For any αϵ(0,d), we construct Cantor sets in Rd of Hausdorff dimension α such that the associated natural measure μ obeys the restriction estimate |f dμ|p≤Cp | f|L(μ) for all p>2d/α. This range is optimal except for the endpoint. This extends the earlier work of Chen, Chen-Seeger, and Shmerkin-Suomala, where a similar result was obtained by different methods for α=d/k with kϵ N. Our proof is based on the decoupling techniques of Bourgain-Demeter and a theorem of Bourgain on the existence of Λ(p) sets.