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

Shmerkin, Pablo

doi:10.4007/annals.2019.189.2.1

Cited by 112 publications

(207 citation statements)

References 51 publications

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“…Finally,

P_{1} false(M_{i, j} false)

corresponds to a non‐principal slice in the product set

π_{m_{1}} false({\tilde{F}}_{ω_{i}} false) \times π_{m_{2}} false({\tilde{E}}_{η_{j}} false)

. By Theorem 1.7, we see that

\bar{{dim}_{B}} (P_{1} (M_{i, j})) ⩽ \max_{false(i, j false) \in false[n_{1} false] \times false[n_{2} false]} \{\frac{\log | Γ_{i} |}{\log m_{1}} + \frac{\log | Λ_{j} |}{\log m_{2}} - 1, 0\} .

In addition,

P_{2} false(M_{i, j} false)

corresponds to a non‐principal slice in the product set

P_{2} false(F false) \times P_{2} false(E false)

, so by Theorem 1.7 (or by the main results of [27] and [26])

\bar{{dim}_{B}} (P_{2} (M_{i, j})) ⩽ \max \{prefixdim P_{2} (F) + prefixdim P_{2} (E) - 1, 0\} .

Since

M \subseteq ⋃ M_{i, j}

(a fini...…”

Section: Proof Of the Main Resultsmentioning

confidence: 91%

“…In addition, P 2 (M i,j ) corresponds to a non-principal slice in the product set P 2 (F ) × P 2 (E), so by Theorem 1.7 (or by the main results of [27] and [26]) Let μ ∈ P (F ) and ν ∈ P (E) be self-affine measures that satisfy the conditions of Theorem 1.5 part (1). Let g : R 2 → R 2 be an affine map such that its linear part is given by a diagonal matrix.…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

“…Recently, two landmark papers have proven, simultaneously and independently, this conjecture to be correct: One of them, by Shmerkin [26], proved it by computing the

L^{q}

dimensions of all the projections of products of invariant Cantor–Lebesgue measures. The second approach, by Wu [27], followed initially along the original idea of Furstenberg by constructing a stationary distribution on the space of measures on slices of

X \times Y

.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 1.4 is related to a long line of research about intersections of Cantor sets. Notable related works include, for example, those of Shmerkin [26] and Wu [27] that proved Conjecture 1.1, the work of Feng, Huang and Rao [12] and the work of Elekes, Keleti and Máthé [7]. Also, it is quite easy to see that the assumption that the carpets are incommensurable cannot be lifted from Theorem 1.4.…”

Section: Introductionmentioning

confidence: 99%

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Slicing theorems and rigidity phenomena for self‐affine carpets

Algom

2020

Proc. London Math. Soc.

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Let F be a Bedford–McMullen carpet defined by independent exponents. We prove that dim¯Bfalse(ℓ∩Ffalse)⩽maxfalse{dim∗F−1,0false} for all lines ℓ not parallel to the principal axes, where dim∗ is Furstenberg's star dimension (maximal dimension of a microset). We also prove several rigidity results for incommensurable Bedford–McMullen carpets, that is, carpets F and E such that all defining exponents are independent: Assuming various conditions, we find bounds on the dimension of the intersection of such carpets, show that self‐affine measures on them are mutually singular, and prove that they do not embed affinely into each other. We obtain these results as an application of a slicing theorem for products of certain Cantor sets. This theorem is a generalization of the results of Shmerkin [Ann. of Math. (2) 189 (2019) 319–391] and Wu [Ann. of Math. (2) 189 (2019) 707–751], which proved Furstenberg's slicing conjecture [Problems in analysis (ed. R. C. Gunning; Princeton University Press, Princeton, NJ, 1970) 41–59].

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“…Finally,

P_{1} false(M_{i, j} false)

corresponds to a non‐principal slice in the product set

π_{m_{1}} false({\tilde{F}}_{ω_{i}} false) \times π_{m_{2}} false({\tilde{E}}_{η_{j}} false)

. By Theorem 1.7, we see that

\bar{{dim}_{B}} (P_{1} (M_{i, j})) ⩽ \max_{false(i, j false) \in false[n_{1} false] \times false[n_{2} false]} \{\frac{\log | Γ_{i} |}{\log m_{1}} + \frac{\log | Λ_{j} |}{\log m_{2}} - 1, 0\} .

In addition,

P_{2} false(M_{i, j} false)

corresponds to a non‐principal slice in the product set

P_{2} false(F false) \times P_{2} false(E false)

, so by Theorem 1.7 (or by the main results of [27] and [26])

\bar{{dim}_{B}} (P_{2} (M_{i, j})) ⩽ \max \{prefixdim P_{2} (F) + prefixdim P_{2} (E) - 1, 0\} .

Since

M \subseteq ⋃ M_{i, j}

(a fini...…”

Section: Proof Of the Main Resultsmentioning

confidence: 91%

Section: Proof Of Theorem 12mentioning

confidence: 99%

“…Recently, two landmark papers have proven, simultaneously and independently, this conjecture to be correct: One of them, by Shmerkin [26], proved it by computing the

L^{q}

X \times Y

.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Slicing theorems and rigidity phenomena for self‐affine carpets

Algom

2020

Proc. London Math. Soc.

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“…By Lemma 2.2 the IFS {ϕ w } w∈Λ m has exponential separation. Thus from [Sh,Theorem 6.6] it follows that the L q dimension of Πν is equal to min{ τ q−1 , 1}. Write…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

Dimension of ergodic measures projected onto self‐similar sets with overlaps

Jordan

Rapaport

2020

Proc. London Math. Soc.

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For self-similar sets on R satisfying the exponential separation condition we show that the natural projections of shift invariant ergodic measures is equal to min{1, h −χ }, where h and χ are the entropy and Lyapunov exponent respectively. The proof relies on Shmerkin's recent result on the L q dimension of self-similar measures. We also use the same method to give results on convolutions and orthogonal projections of ergodic measures projected onto self-similar sets.

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$$L^q$$ Dimensions of Self-similar Measures and Applications: A Survey

Shmerkin

2019

New Trends in Applied Harmonic Analysis, Volume 2

Self Cite

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We present a self-contained proof of a formula for the L q dimensions of self-similar measures on the real line under exponential separation (up to the proof of an inverse theorem for the L q norm of convolutions). This is a special case of a more general result of the author from [Shmerkin, Pablo. On Furstenberg's intersection conjecture, self-similar measures, and the L q norms of convolutions. Ann. of Math., 2019], and one of the goals of this survey is to present the ideas in a simpler, but important, setting. We also review some applications of the main result to the study of Bernoulli convolutions and intersections of self-similar Cantor sets..2010 Mathematics Subject Classification. Primary: 28A75, 28A80.

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

Cited by 112 publications

References 51 publications

Slicing theorems and rigidity phenomena for self‐affine carpets

Slicing theorems and rigidity phenomena for self‐affine carpets

Dimension of ergodic measures projected onto self‐similar sets with overlaps

$$L^q$$ Dimensions of Self-similar Measures and Applications: A Survey

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