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

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