Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions

Abstract

Erdős (1939, 1940) studied the distribution νλ of the random series P∞ 0 ±λn, and showed that νλ is singular for infinitely many λ ∈ (1/2, 1), and absolutely continuous for a.e. λ in a small interval (1 − δ, 1). Solomyak (1995) proved a conjecture made by Garsia (1962) that νλ is absolutely continuous for a.e. λ ∈ (1/2, 1). In order to sharpen this result, we have developed a general method that can be used to estimate the Hausdorff dimension of exceptional parameters in several contexts. In particular, we prove: • For any λ0 > 1/2, the set of λ ∈ [λ0, 1) such that νλ is singular has Hausdorff dimension strictly less than 1. • For any Borel set A ⊂ Rd with Hausdorff dimension dim A > (d + 1)/2, there are points x ∈ A such that the pinned distance set {|x− y| : y ∈ A} has positive Lebesgue measure. Moreover, the set of x where this fails has Hausdorff dimension at most d + 1− dim A. • Let Kλ denote the middle-α Cantor set for α = 1 − 2λ and let K ⊂ R be any compact set. Peres and Solomyak (1998) showed that for a.e. λ ∈ (λ0, 1/2) such that dim K + dimKλ > 1, the sum K + Kλ has positive length; we show that the set of exceptional λ in this statement has Hausdorff dimension at most 2− dim K − dimKλ0 . • For any Borel set E ⊂ Rd with dim E > 2, almost all orthogonal projections of E onto lines through the origin have nonempty interior, and the exceptional set of lines where this fails has dimension at most d + 1− dim E. • If μ is a Borel probability measure on Rd with correlation dimension greater than m + 2γ, then for a “prevalent” set of C1 maps f : Rd → Rm (in the sense described by Hunt, Sauer and Yorke (1992)), the image of μ under f has a density with at least γ fractional derivatives in L2(Rm).