Some Problems on the Boundary of Fractal Geometry and Additive Combinatorics

Abstract

This paper is an exposition, with some new applications, of our results from [5,6] on the growth of entropy of convolutions. We explain the main result on R, and derive, via a linearization argument, an analogous result for the action of the affine group on R. We also develop versions of the results for entropy dimension and Hausdorff dimension. The method is applied to two problems on the border of fractal geometry and additive combinatorics. First, we consider attractors X of compact families Φ of similarities of R. We conjecture that if Φ is uncountable and X is not a singleton (equivalently, Φ is not contained in a 1-parameter semigroup) then dim X = 1. We show that this would follow from the classical overlaps conjecture for self-similar sets, and unconditionally we show that if X is not a point and dim Φ > 0 then dim X = 1. Second, we study a problem due to Shmerkin and Keleti, who have asked how small a set ∅ = Y ⊆ R can be if at every point it contains a scaled copy of the middle-third Cantor set K. Such a set must have dimension at least dim K and we show that its dimension is at least dim K + δ for some constant δ > 0.

Small Union with Large Set of Centers

“…One can also ask if at least we can guarantee that the union of scaled copies of a Cantor set C around every point of R has Hausdorff dimension strictly larger than the Hausdorff dimension of C. Very recently M. Hochman [9] gave an affirmative answer to this question for porous Cantor sets. (Here a set is called porous if there exist c > 0 and r 0 > 0 such that every interval of length r < r 0 contains an interval of length cr disjoint to the set.)…”

“…Theorem 3.3. (Hochman (2016) [9]) Let S ⊂ R be a compact set with dim H S > 0, C ⊂ R be a porous Cantor set. If B ⊂ R contains a scaled copy rC + x of C for every x ∈ S then dim H B > dim H C + δ, where δ > 0 depends only on dim S and dim C.…”

“…In case of Theorem 3.3 see the explanation in [9] before the proof the theorem. In case of Corollary 3.5, as Máthé pointed out, if we consider upper box dimension instead of Hausdorff dimension then Corollary 3.5 would follow very easily even if we assume only that C has at least two points.…”

“…Recall from Section 3 that by the theorem of Laba and Pramanik [13] there exist Cantor sets for which such a B must have positive Lebesgue measure, by the result of Máthé [16] there exist Cantor sets for which such a B can have zero measure, and by the results of Hochman [9] and Bourgain [2] for any porous Cantor set C with dim H C > 0 such a B must have Hausdorff dimension strictly larger than dim H C and at least 1/2. Theorem 6.1 shows that in some cases we can get sets with smaller Hausdorff dimension if we allow rotation but we do not know if smaller box or smaller packing dimension can be also obtained by allowing rotated squares or cubes.…”

“…In this case there are four results: Laba and Pramanik [13] constructed Cantor sets T ⊂ [1,2] for which the Lebesgue measure of B must be positive whenever S has positive Lebesgue measure; Máthé [16] constructed Cantor sets T for which this is false; Hochman [9] proved that if dim H S > 0 then dim H B > dim H C + δ for any porous Cantor set T , where δ > 0 depends only on dim H T and dim H S; and Máthé noticed that it is a consequence of a recent projection theorem of Bourgain [2] that dim H C > 0 implies dim H B ≥ dim H S 2 . In Section 4 we present the results of Nagy, Shmerkin and the author [11] about the case when n = 2 and T is the boundary or the set of vertices of a fixed axis parallel square centered at the origin.…”