We study the relationship between the sizes of two sets B, S ⊂ R 2 when B contains either the whole boundary, or the four vertices, of a square with axes-parallel sides and center in every point of S, where size refers to one of cardinality, Hausdorff dimension, packing dimension, or upper or lower box dimension. Perhaps surprinsingly, the results vary depending on the notion of size under consideration. For example, we construct a compact set B of Hausdorff dimension 1 which contains the boundary of an axes-parallel square with center in every point [0, 1] 2 , but prove that such a B must have packing and lower box dimension at least 7 4 , and show by example that this is sharp. For more general sets of centers, the answers for packing and box counting dimensions also differ. These problems are inspired by the analogous problems for circles that were investigated by Bourgain, Marstrand and Wolff, among others.

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 + 3s(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 multiscale 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.2010 Mathematics Subject Classification. Primary: 28A75, 28A80; Secondary: 26A16, 49Q15.

We prove that for any 1 ≤ k < n and s ≤ 1, the union of any nonempty s-Hausdorff dimensional family of k-dimensional affine subspaces of R n has Hausdorff dimension k + s. More generally, we show that for any 0 < α ≤ k, if B ⊂ R n and E is a nonempty collection of k-dimensional affine subspaces of R n such that every P ∈ E intersects B in a set of Hausdorff dimension at least α, then dim B ≥ 2α − k + min(dim E, 1), where dim denotes the Hausdorff dimension. As a consequence, we generalize the well-known Furstenberg-type estimate that every α-Furstenberg set has Hausdorff dimension at least 2α; we strengthen a theorem of Falconer and Mattila [5]; and we show that for any 0 ≤ k < n, if a set A ⊂ R n contains the k-skeleton of a rotated unit cube around every point of R n , or if A contains a k-dimensional affine subspace at a fixed positive distance from every point of R n , then the Hausdorff dimension of A is at least k + 1.

We construct a compact subset of R with Hausdorff dimension 1 that intersects each of its non-identical translates in at most one point. Moreover, one can make the set to be linearly independent over the rationals.

Our aim is to find the minimal Hausdorff dimension of the union of scaled and/or rotated copies of the k-skeleton of a fixed polytope centered at the points of a given set. For many of these problems, we show that a typical arrangement in the sense of Baire category gives minimal Hausdorff dimension. In particular, this proves a conjecture of R. Thornton.Our results also show that Nikodym sets are typical among all sets which contain, for every x ∈ R n , a punctured hyperplane H \ {x} through x. With similar methods we also construct a Borel subset of R n of Lebesgue measure zero containing a hyperplane at every positive distance from every point.2010 Mathematics Subject Classification. 28A78, 54E52.