Convolutions of cantor measures without resonance

Abstract

Abstract. Denote by µ a the distribution of the random sum (1 − a) ∞ j=0 ω j a j , where P(ω j = 0) = P(ω j = 1) = 1/2 and all the choices are independent. For 0 < a < 1/2, the measure µ a is supported on C a , the central Cantor set obtained by starting with the closed united interval, removing an open central interval of length (1 − 2a), and iterating this process inductively on each of the remaining intervals. We investigate the convolutions µ a * (µ b • S −1 λ ), where S λ (x) = λx is a rescaling map. We prove that if the ratio log b/ log a is irrational and λ = 0, thenwhere D denotes any of correlation, Hausdorff or packing dimension of a measure.We also show that, perhaps surprisingly, for uncountably many values of λ the convolution µ 1/4 * (µ 1/3 •S −1 λ ) is a singular measure, although dim H (C

A family of smooth Cantor sets

“…Lau et al ([FLN00], [HL01]) studied the multifractal structure of the m-th convolution of the measure µ 1/3 , which is singular by the above argument. Nazarov et al [NPS09] determined that the correlation dimension of µ r * µ s is min d r + d s , 1 whenever log r/ log s is irrational. Shmerkin informed us that in a joint work with Hochman [HS09] they generalize the work on sums of Cantor sets [PS09] and their methods imply that the dimension of the convolution H p * H p is min dim C p + dim C p , 1 whenever p/p is irrational.…”

On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions

“…The second part of the following proposition can be used to give another (though closely related) proof of Proposition 4.6, and was obtained in [40,35] in special cases. The first part is proved in a similar way, relying on Lemma 4.12.…”

“…This shows that µ x,n → µ x weakly. Although in different language, this decomposition of µ x can be traced back to Furstenberg [20], and was also used more explicitly in [35] to study the L 2 dimensions of µ x .…”

“…[33, Proposition 3.2] for the proofs. The next proposition introduces a sub-multiplicative cocycle (which was first used in [35], in a special case) that will play a crucial rôle in the proof of Theorem 1.11. Let us define the following sequence of functions, parametrized by q ∈ [1, ∞):…”

Singularity versus exact overlaps for self-similar measures

“…For an i ∈ Σ we define the natural projection Π α (i) as in (6). Clearly, The fact that Property P5A holds for the special case in the example was proved in [4, p.216].…”