Tuomas Orponen

On the distance sets of self-similar sets

We show that if K is a self-similar set in the plane with positive length, then the distance set of K has Hausdorff dimension one.

On the distance sets of Ahlfors–David regular sets

Abstract I prove that if ∅ ≠ K ⊂ R 2 is a compact s-Ahlfors–David regular set with s ≥ 1 , then dim p ⁡ D ( K ) = 1 , where D ( K ) : = { | x − y | : x , y ∈ K } is the distance set of K, and dim p stands for packing dimension. The same proof strategy applies to other problems of similar nature. For instance, one can show that if ∅ ≠ K ⊂ R 2 is a compact s-Ahlfors–David regular set with s ≥ 1 , then there exists a point x 0 ∈ K such that dim p ⁡ K ⋅ ( K − x 0 ) = 1 .

On restricted families of projections in ℝ3

We study projections onto non-degenerate one-dimensional families of lines and planes in R 3 . Using the classical potential theoretic approach of R. Kaufman, one can show that the Hausdorff dimension of at most 12 -dimensional sets [Math Processing Error] is typically preserved under one-dimensional families of projections onto lines. We improve the result by an e , proving that if [Math Processing Error], then the packing dimension of the projections is almost surely at least [Math Processing Error]. For projections onto planes, we obtain a similar bound, with the threshold 12 replaced by 1 . In the special case of self-similar sets [Math Processing Error] without rotations, we obtain a full Marstrand-type projection theorem for 1-parameter families of projections onto lines. The [Math Processing Error] case of the result follows from recent work of M. Hochman, but the [Math Processing Error] part is new: with this assumption, we prove that the projections have positive length almost surely.

The Assouad dimensions of projections of planar sets

The first named author is supported by a Leverhulme Trust Research Fellowship and the second named author is supported by the Academy of Finland through the grant Restricted families of projections and connections to Kakeya type problems, grant number 274512.

Projections of planar sets in well-separated directions

Abstract This paper contains two new projection theorems in the plane. First, let K ⊂ B ( 0 , 1 ) ⊂ R 2 be a set with H ∞ 1 ( K ) ∼ 1 , and write π e ( K ) for the orthogonal projection of K into the line spanned by e ∈ S 1 . For 1 / 2 ≤ s 1 , write E s : = { e : N ( π e ( K ) , δ ) ≤ δ − s } , where N ( A , r ) is the r -covering number of the set A . It is well-known – and essentially due to R. Kaufman – that N ( E s , δ ) ⪅ δ − s . Using the polynomial method, I prove that N ( E s , r ) ⪅ min ⁡ { δ − s ( δ r ) 1 / 2 , r − 1 } , δ ≤ r ≤ 1 . I construct examples showing that the exponents in the bound are sharp for δ ≤ r ≤ δ s . The second theorem concerns projections of 1-Ahlfors–David regular sets. Let A ≥ 1 and 1 / 2 ≤ s 1 be given. I prove that, for p = p ( A , s ) ∈ N large enough, the finite set of unit vectors S p : = { e 2 π i k / p : 0 ≤ k p } has the following property. If K ⊂ B ( 0 , 1 ) is non-empty and 1-Ahlfors–David regular with regularity constant at most A , then 1 p ∑ e ∈ S p N ( π e ( K ) , δ ) ≥ δ − s for all small enough δ > 0 . In particular, dim ‾ B π e ( K ) ≥ s for some e ∈ S p .

Constancy results for special families of projections

Let { = V × ℝ l : V ∈ G(n−l,m−l )} be the family of m -dimensional subspaces of ℝ n containing {0} × ℝ l , and let : ℝ n → be the orthogonal projection onto . We prove that the mapping V ↦ Dim ( B ) is almost surely constant for any analytic set B ⊂ ℝ n , where Dim denotes either Hausdorff or packing dimension.

Tangent measures of non-doubling measures

We construct a non-doubling measure on the real line, all tangent measures of which are equivalent to Lebesgue measure.

A sharp exceptional set estimate for visibility

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RADIAL PROJECTIONS OF RECTIFIABLE SETS

B \subset \mathbb{R}^{n}

On the conformal dimension of product measures

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