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1. CJM 2013 (vol 66 pp. 303)
Haar Null Sets and the Consistent Reflection of Non-meagreness A subset $X$ of a Polish group $G$ is called Haar null if there exists
a Borel set $B \supset X$ and Borel probability measure $\mu$ on $G$ such that
$\mu(gBh)=0$ for every $g,h \in G$.
We prove that there exist a set $X \subset \mathbb R$ that is not Lebesgue null and a
Borel probability measure $\mu$ such that $\mu(X + t) = 0$ for every $t \in
\mathbb R$.
This answers a question from David Fremlin's problem list by showing
that one cannot simplify the definition of a Haar null set by leaving out the
Borel set $B$. (The answer was already known assuming the Continuum
Hypothesis.)
This result motivates the following Baire category analogue. It is consistent
with $ZFC$ that there exist an abelian Polish group $G$ and a Cantor
set $C \subset G$ such that for every non-meagre set $X \subset G$ there exists a $t
\in G$ such that $C \cap (X + t)$ is relatively non-meagre in $C$. This
essentially generalises results of BartoszyÅski and Burke-Miller.
Keywords:Haar null, Christensen, non-locally compact Polish group, packing dimension, Problem FC on Fremlin's list, forcing, generic real Categories:28C10, 03E35, 03E17, , , , , 22C05, 28A78 |
2. CJM 2010 (vol 62 pp. 543)
More Variations on the SierpiÅski Sieve This paper answers a question of Broomhead, Montaldi and Sidorov about the existence of gaskets of a particular type related to the SierpiÅski sieve. These gaskets are given by iterated function systems that do not satisfy the open set condition. We use the methods of Ngai and Wang to compute the dimension of these gaskets.
Categories:28A80, 28A78, 11R06 |
3. CJM 2004 (vol 56 pp. 115)
Estimates of Hausdorff Dimension for the Non-Wandering Set of an Open Planar Billiard The billiard flow in the plane has a simple geometric definition; the
movement along straight lines of points except where elastic
reflections are made with the boundary of the billiard domain. We
consider a class of open billiards, where the billiard domain is
unbounded, and the boundary is that of a finite number of strictly
convex obstacles. We estimate the Hausdorff dimension of the
nonwandering set $M_0$ of the discrete time billiard ball map, which
is known to be a Cantor set and the largest invariant set. Under
certain conditions on the obstacles, we use a well-known coding of
$M_0$ \cite{Morita} and estimates using convex fronts related to the
derivative of the billiard ball map \cite{StAsy} to estimate the
Hausdorff dimension of local unstable sets. Consideration of the
local product structure then yields the desired estimates, which
provide asymptotic bounds on the Hausdorff dimension's convergence to
zero as the obstacles are separated.
Categories:37D50, 37C45;, 28A78 |
4. CJM 2002 (vol 54 pp. 1280)
Besov Spaces and Hausdorff Dimension For Some Carnot-CarathÃ©odory Metric Spaces We regard a system of left invariant vector fields $\mathcal{X}=\{X_1,\dots,X_k\}$
satisfying the H\"ormander condition and the related Carnot-Carath\'eodory metric on a
unimodular Lie group $G$. We define Besov spaces corresponding to the sub-Laplacian
$\Delta=\sum X_i^2$ both with positive and negative smoothness. The atomic
decomposition of the spaces is given. In consequence we get the distributional
characterization of the Hausdorff dimension of Borel subsets with the Haar measure
zero.
Keywords:Besov spaces, sub-elliptic operators, Carnot-CarathÃ©odory metric, Hausdorff dimension Categories:46E35, 43A15, 28A78 |
5. CJM 1999 (vol 51 pp. 1073)
The Hausdorff and Packing Dimensions of Some Sets Related to Sierpi\'nski Carpets The Sierpi\'nski carpets first considered by C.~McMullen and later
studied by Y.~Peres are modified by insisting that the allowed
digits in the expansions occur with prescribed frequencies. This
paper (i)~~calculates the Hausdorff, box (or Minkowski), and
packing dimensions of the modified Sierpi\'nski carpets and
(ii)~~shows that for these sets the Hausdorff and packing measures
in their dimension are never zero and gives necessary and
sufficient conditions for these measures to be infinite.
Categories:28A78, 28A80 |