Absolute continuity of complex Bernoulli convolutions

Shmerkin, Pablo; Solomyak, Boris

doi:10.1017/s0305004116000335

Cited by 9 publications

(7 citation statements)

References 28 publications

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“…If the planar self-similar measure µ has power Fourier decay and α/π is irrational, then the proof of the Corollary 9.3 together with the above observations show that P x µ has an L q density for all x ∈ S 1 , whenever ∆ q q < λ (q−1) . Although we know of no explicit example of such measure µ, in parameter space power Fourier decay occurs outside of very small exceptional sets; see [45,Theorem D].…”

Section: Absolute Continuity and L Q Densitiesmentioning

confidence: 99%

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

2019

Self Cite

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We study a class of measures on the real line with a kind of self-similar structure, which we call dynamically driven self-similar measures, and contain proper self-similar measures such as Bernoulli convolutions as special cases. Our main result gives an expression for the L q dimensions of such dynamically driven self-similar measures, under certain conditions. As an application, we settle Furstenberg's longstanding conjecture on the dimension of the intersections of ×p and ×q-invariant sets. Among several other applications, we also show that Bernoulli convolutions have an L q density for all finite q, outside of a zero-dimensional set of exceptions.The proof of the main result is inspired by M. Hochman's approach to the dimensions of self-similar measures and his inverse theorem for entropy. Our method can be seen as an extension of Hochman's theory from entropy to L q norms, and likewise relies on an inverse theorem for the decay of L q norms of discrete measures under convolution. This central piece of our approach may be of independent interest, and is an application of well-known methods and results in additive combinatorics: the asymmetric version of the Balog-Szemerédi-Gowers Theorem due to Tao-Vu, and some constructions of Bourgain.

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Section: Absolute Continuity and L Q Densitiesmentioning

confidence: 99%

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

2019

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“…Part (1) follows almost directly from results which are due to Shmerkin and Solomyak [SS1] and [SS2] and Shmerkin [Sh1] and [Sh2]. Our main contribution is the derivation of parts (2) and (3).…”

Section: Introductionmentioning

confidence: 69%

On self-similar measures with absolutely continuous projections and dimension conservation in each direction

Rapaport¹

2019

Ergod. Th. Dynam. Sys.

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Relying on results due to Shmerkin and Solomyak, we show that outside a 0-dimensional set of parameters, for every planar homogeneous selfsimilar measure ν, with strong separation, dense rotations and dimension greater than 1, there exists q > 1 such that {Pzν} z∈S ⊂ L q (R). Here S is the unit circle and Pzw = z, w for w ∈ R 2 . We then study such measures. For instance, we show that ν is dimension conserving in each direction and that the map z → Pzν is continuous with respect to the weak topology of L q (R).

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“…This argument yields not only that ν λ is absolutely continuous (if λ is outside the exceptional set) but also that it has some fractional derivatives in L p for some p > 1 (depending on λ). Shmerkin and Solomyak [56], [57] extended these ideas to more general self-similar measures.…”

Section: 2mentioning

confidence: 99%