Translates of compact sets
and combinatorics of fractal dimensions
Ondřej Zindulka
Czech Technical University Prague
September 14, 2012
Ondřej Zindulka
Translates of compact sets and combinatorics of fractal dimensions
A property of Cantor set
Theorem (Gruenhage 2002)
Let C ⊆ R be the Cantor set and T ⊆ R. If |T | < c, then C + T 6= R.
Question (Gruenhage 2002)
Is there K ⊆ R null compact such that C + T = R for some T ∈ [R]<c ?
Question (Mauldin)
Is there K ⊆ R compact with dimH K < 1 such that C + T = R for some
T ∈ [R]<c ?
Ondřej Zindulka
Translates of compact sets and combinatorics of fractal dimensions
Thin sets
Definition
A set X ⊆ R is thin if ∀T ∈ [R]<c X + T 6= R.
Ondřej Zindulka
Translates of compact sets and combinatorics of fractal dimensions
Consistency results
Theorem
con(ZFC + X is thin ⇔ X is Lebesgue null)
con(ZFC + ∃X compact null that is not thin)
con(ZFC + thin sets form a σ-ideal)
con(ZFC + thin sets do not form an ideal)
Lemma (Steinhaus)
If A, B are not null and B is compact, then A + B contains an interval.
Ondřej Zindulka
Translates of compact sets and combinatorics of fractal dimensions
Witness
Definition
A compact set C is a witness to X if ∀t ∈ R |(X + t) ∩ C| < c
Proposition
If X has a witness, then X is thin.
Theorem (Elekes and Steprans 2004)
There is E compact null that does not have a witness and
E is thin under CH,
E is not thin in the Sacks model.
Question (Oz)
Is it consitent that every thin compact set has a witness?
Ondřej Zindulka
Translates of compact sets and combinatorics of fractal dimensions
Sacks model
Question (Oz)
Is it consitent that every thin compact set has a witness?
Definition (M. Hrušák, Oz 2011)
A Borel ideal J is a Borel witness to X if ∀t ∈ R X + t ∈ J and
ZFC ` cov(J ) = c.
Proposition
If X has a Borel witness, then X is thin.
Theorem (M. Hrušák, Oz 2011)
In the Sacks model, a compact set has a Borel witness iff it is thin.
Ondřej Zindulka
Translates of compact sets and combinatorics of fractal dimensions
Powerful Hausdorff dimension
Definition (Oz 2005)
Let E be a set in a metric space. Define
dimπH E = sup
n∈ω
dimH E n
dimH E n
= lim
.
n∈ω
n
n
Theorem (Oz 2011)
Let X, Y be metric spaces, ϕ : X × R<ω → Y a σ-Lipschitz map.
Let T ⊆ R<ω , |T | < c. If A ⊆∗ ϕ(X × T ) is analytic, then
dimπH A 6 dimπH X.
Corollary (Oz 2011)
There is B ⊆ Y such that
If A ⊆∗ B is analytic, then dimπH A 6 dimπH X,
ϕ(X × {t}) ⊆∗ B for all t ∈ R<ω .
Ondřej Zindulka
Translates of compact sets and combinatorics of fractal dimensions
Powerful Hausdorff dimension vs. lower packing dimension
Theorem (Oz 2011)
Let X, Y be metric spaces, ϕ : X × R<ω → Y a σ-Lipschitz map.
Let T ⊆ R<ω , |T | < c. If A ⊆∗ ϕ(X × T ) is analytic, then
dimP A 6 dimP X.
Question (Oz)
Is there X ⊆ R compact such that dimπH X < dimP X?
Is there X ⊆ R compact such that dimP X < dimπH X?
Ondřej Zindulka
Translates of compact sets and combinatorics of fractal dimensions
Precipitous nets
Definition
Let X be a set in a metric space.
A finite family of subsets of X is termed a partial cover of X.
diam E = sup{diam E : E ∈ E}.
A sequence E = {En : n ∈ ω} is termed a net.
b = {S En : n ∈ ω}.
E
Definition
A net E is precipitous if
log n|En |
−→ 0.
log diam En
Ondřej Zindulka
Translates of compact sets and combinatorics of fractal dimensions
Combinatorics of covers and dimensions
Definition
A cover {Un : n ∈ ω} of a set X is
a λ-cover if ∀x ∈ X ∃∞ n x ∈ Un ,
a γ-cover if ∀x ∈ X ∀∞ n x ∈ Un ,
an ω-cover if ∀F ∈ [X]<ω ∃∞ n x ∈ Un ,
a γ-groupable
S cover if there is a partition I0 , I1 , . . . of ω into finite sets
such that { n∈Ik Un : k ∈ ω} is a γ-cover.
Theorem (Oz 2011)
b is a λ-cover.
dimH X = 0 ⇔ ∃ precipitous net E such that E
b is a γ-cover.
dimP X = 0 ⇔ ∃ precipitous net E such that E
−−→
b is a γ-groupable cover.
dimH X = 0 ⇔ ∃ precipitous net E such that E
b is a ω-cover.
dimπH X = 0 ⇔ ∃ precipitous net E such that E
Ondřej Zindulka
Translates of compact sets and combinatorics of fractal dimensions
A metric on 2ω
Definition
For p ∈ 2<ω , x, y ∈ 2ω
κ(p) = min{n : ∀i ∈ [n, |p|) p(i) = 0}
1
χ(p) =
(|p| + 1)κ(p)!
ρ(x, y) = χ(x ∧ y)
X = (2ω , ρ)
Definition
En = {[p] : p ∈ 2n+1 , κ(p) = n + 1}
Ondřej Zindulka
Translates of compact sets and combinatorics of fractal dimensions
Dimensions in X
Lemma
∀x ∈ 2ω {n : x ∈ En } = x
Proposition
dimH X = 0
If F ⊆ X is centered, then dimπH F = 0
If F ⊆ X is nonmeager, then dimP F > 1
Theorem
If F ⊆ X is an ultrafilter, then dimπH F = 0 < 1 6 dimP F .
Ondřej Zindulka
Translates of compact sets and combinatorics of fractal dimensions
Rich trees
Definition
A tree T ⊆ 2<ω is rich if ∀p ∈ T ∃q ⊇ p Sq ⊆ T.
Theorem (M. Hrušák, Oz 2011)
There is a rich tree which set of branches is centered.
Corollary (M. Hrušák, Oz 2011)
There is a perfect set F ⊆ X such that dimπH F = 0 < 1 6 dimP F .
Ondřej Zindulka
Translates of compact sets and combinatorics of fractal dimensions
The interval
Lemma
There is a Lipschitz map ϕ : X → [0, 1] such that ϕ−1 is
1
-Hölder.
3
Corollary
There is a compact set E ⊆ R such that dimπH E = 0 <
1
3
6 dimP E.
Lemma (Oz 2011)
dimP X × Y > dimP X + dimP Y
Lemma (Falconer and Howroyd 1995)
Let n > m > 1. Let X ⊆ Rn be a Borel set. There is a projection π on Rm
such that
dimP X
dimP π(X) >
.
1
1+ m
− n1 dimP X
Ondřej Zindulka
Translates of compact sets and combinatorics of fractal dimensions
Conclusion
Theorem (M. Hrušák, Oz 2011)
There is a compact set E ⊆ R such that dimπH E = 0 < 1 = dimP E.
Theorem (M. Hrušák, Oz 2011)
There is a compact set E ⊆ R such that dimP E = 0 < 1 = dimπH E.
Ondřej Zindulka
Translates of compact sets and combinatorics of fractal dimensions