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We provide several new answers on the question: how do radial projections distort the dimension of planar sets? Let be non-empty Borel sets. If X is not contained in any line, we prove that If dimHY>1, we have the following improved lower bound: Our results solve conjectures of Lund-Thang-Huong, Liu, and the first author. Another corollary is the following continuum version of Beck's theorem in combinatorial geometry: if is a Borel set with the property that dimH(X ∖ ℓ)=dimHX for all lines , then the line set spanned by X has Hausdorff dimension at least min{2dimHX,2}. While the results above concern , we also derive some counterparts in by means of integralgeometric considerations. The proofs are based on an ϵ-improvement in the Furstenberg set problem, due to the two first authors, a bootstrapping scheme introduced by the second and third author, and a new planar incidence estimate due to Fu and Ren. [ABSTRACT FROM AUTHOR]

We prove a version of Bourgain's projection theorem for parametrized families of C2 maps, which refines the original statement even in the linear case by requiring non-concentration only at a single natural scale. As one application, we show that if A is a Borel set of Hausdorff dimension close to 1 in R² or close to 3=2 in R³, then for y ∈ A outside of a very sparse set, the pinned distance set {|x - y|: x ∈ A} has Hausdorff dimension at least 1=2 C c, where c is universal. Furthermore, the same holds if the distances are taken with respect to a C² norm of positive Gaussian curvature. As further applications, we obtain new bounds on the dimensions of spherical projections, and an improvement over the trivial estimate for incidences between ı-balls and δ-neighborhoods of curves in the plane, under fairly general assumptions. The proofs depend on a new multiscale decomposition of measures into "Frostman pieces" that may be of independent interest. [ABSTRACT FROM AUTHOR]

We show that given α∈(0,1) there is a constant c=c(α)>0 such that any planar (α,2α)-Furstenberg set has Hausdorff dimension at least 2α+c. This improves several previous bounds, in particular extending a result of Katz-Tao and Bourgain. We follow the Katz-Tao approach with suitable changes, along the way clarifying, simplifying and/or quantifying many of the steps. [ABSTRACT FROM AUTHOR]

We obtain new lower bounds on the Hausdorff dimension of distance sets and pinned distance sets of planar Borel sets of dimension slightly larger than 1, improving recent estimates of Keleti and Shmerkin, and of Liu in this regime. In particular, we prove that if dimH (A) > 1, then the set of distances spanned by points of A has Hausdorff dimension at least 40/57 > 0.7 and there are many y ∈ A such that the pinned distance set {\x - y\: x ∈ A} has Hausdorff dimension at least 29/42 and lower box-counting dimension at least 40/57. We use the approach and many results from the earlier work of Keleti and Shmerkin, but incorporate estimates from the recent work of Guth, Iosevich, Ou and Wang as additional input. [ABSTRACT FROM AUTHOR]

We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if A ⊂ R 2 is a Borel set of Hausdorff dimension s > 1 , then its distance set has Hausdorff dimension at least 37 / 54 ≈ 0.685 . Moreover, if s ∈ (1 , 3 / 2 ] , then outside of a set of exceptional y of Hausdorff dimension at most 1, the pinned distance set { | x - y | : x ∈ A } has Hausdorff dimension ≥ 2 3 s and packing dimension at least 1 4 (1 + s + 3 s (2 - s)) ≥ 0.933 . These estimates improve upon the existing ones by Bourgain, Wolff, Peres–Schlag and Iosevich–Liu for sets of Hausdorff dimension > 1 . Our proof uses a multi-scale decomposition of measures in which, unlike previous works, we are able to choose the scales subject to certain constrains. This leads to a combinatorial problem, which is a key new ingredient of our approach, and which we solve completely by optimizing certain variation of Lipschitz functions. [ABSTRACT FROM AUTHOR]

Under mild conditions we show that the affinity dimension of a planar self-affine set is equal to the supremum of the Lyapunov dimensions of self-affine measures supported on self-affine proper subsets of the original set. These self-affine subsets may be chosen so as to have stronger separation properties and in such a way that the linear parts of their affinities are positive matrices. Combining this result with some recent breakthroughs in the study of self-affine measures and their associated Furstenberg measures, we obtain new criteria under which the Hausdorff dimension of a self-affine set equals its affinity dimension. For example, applying recent results of B'ar'any, Hochman- Solomyak, and Rapaport, we provide many new explicit examples of self-affine sets whose Hausdorff dimension equals its affinity dimension, and for which the linear parts do not satisfy any positivity or domination assumptions. [ABSTRACT FROM AUTHOR]

We prove that complex Bernoulli convolutions are absolutely continuous in the supercritical parameter region, outside of an exceptional set of parameters of zero Hausdorff dimension. Similar results are also obtained in the biased case, and for other parametrised families of self-similar sets and measures in the complex plane, extending earlier results. [ABSTRACT FROM PUBLISHER]

We prove that the set of exceptional $${\lambda\in (1/2,1)}$$ such that the associated Bernoulli convolution is singular has zero Hausdorff dimension, and likewise for biased Bernoulli convolutions, with the exceptional set independent of the bias. This improves previous results by Erdös, Kahane, Solomyak, Peres and Schlag, and Hochman. A theorem of this kind is also obtained for convolutions of homogeneous self-similar measures. The proofs are very short, and rely on old and new results on the dimensions of self-similar measures and their convolutions, and the decay of their Fourier transform. [ABSTRACT FROM AUTHOR]