Expand all Collapse all | Results 1 - 22 of 22 |
1. CJM Online first
On a sumset conjecture of ErdÅs ErdÅs conjectured that for any set $A\subseteq \mathbb{N}$
with positive
lower asymptotic density, there are infinite sets $B,C\subseteq
\mathbb{N}$
such that $B+C\subseteq A$. We verify ErdÅs' conjecture in
the case that $A$ has Banach density exceeding $\frac{1}{2}$.
As a consequence, we prove that, for $A\subseteq \mathbb{N}$
with
positive Banach density (a much weaker assumption than positive
lower density), we can find infinite $B,C\subseteq \mathbb{N}$
such
that $B+C$ is contained in the union of $A$ and a translate of
$A$. Both of the aforementioned
results are generalized to arbitrary countable
amenable groups. We also provide a positive solution to ErdÅs'
conjecture for subsets of the natural numbers that are pseudorandom.
Keywords:sumsets of integers, asymptotic density, amenable groups, nonstandard analysis Categories:11B05, 11B13, 11P70, 28D15, 37A45 |
2. CJM 2013 (vol 66 pp. 721)
On Whitney-type Characterization of Approximate Differentiability on Metric Measure Spaces We study approximately differentiable functions on metric measure spaces admitting a Cheeger differentiable structure. The main result is a Whitney-type characterization of approximately differentiable functions in this setting.
As an application, we prove a Stepanov-type theorem and consider approximate differentiability of Sobolev, $BV$ and maximal functions.
Keywords:approximate differentiability, metric space, strong measurable differentiable structure, Whitney theorem Categories:26B05, 28A15, 28A75, 46E35 |
3. CJM 2013 (vol 66 pp. 641)
Heat Kernels and Green Functions on Metric Measure Spaces We prove that, in a setting of local Dirichlet forms on metric measure
spaces, a two-sided sub-Gaussian estimate of the heat kernel is equivalent
to the conjunction of the volume doubling propety, the elliptic Harnack
inequality and a certain estimate of the capacity between concentric balls.
The main technical tool is the equivalence between the capacity estimate and
the estimate of a mean exit time in a ball, that uses two-sided estimates of
a Green function in a ball.
Keywords:Dirichlet form, heat kernel, Green function, capacity Categories:35K08, 28A80, 31B05, 35J08, 46E35, 47D07 |
4. 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 |
5. CJM 2011 (vol 63 pp. 648)
Spectral Asymptotics of Laplacians Associated with One-dimensional Iterated Function Systems with Overlaps |
Spectral Asymptotics of Laplacians Associated with One-dimensional Iterated Function Systems with Overlaps We set up a framework for computing the spectral dimension of a class of one-dimensional
self-similar measures that are defined by iterated function systems
with overlaps and satisfy a family of second-order self-similar
identities. As applications of our result we obtain the spectral dimension
of important measures such as the infinite Bernoulli convolution
associated with the golden ratio and convolutions of Cantor-type measures.
The main novelty of our result is that the iterated function systems
we consider are not post-critically finite and do not satisfy the
well-known open set condition.
Keywords:spectral dimension, fractal, Laplacian, self-similar measure, iterated function system with overlaps, second-order self-similar identities Categories:28A80, , , , 35P20, 35J05, 43A05, 47A75 |
6. CJM 2010 (vol 63 pp. 153)
Asymptotics for Functions Associated with Heat Flow on the Sierpinski Carpet
We establish the asymptotic behaviour of the partition function, the
heat content, the integrated eigenvalue counting function, and, for
certain points, the on-diagonal heat kernel of generalized
Sierpinski carpets. For all these functions the leading term is of
the form $x^{\gamma}\phi(\log x)$ for a suitable exponent $\gamma$
and $\phi$ a periodic function. We also discuss similar results for
the heat content of affine nested fractals.
Categories:35K05, 28A80, 35B40, 60J65 |
7. CJM 2010 (vol 62 pp. 1182)
A Fractal Function Related to the John-Nirenberg Inequality for $Q_{\alpha}({\mathbb R^n})$
A borderline case function $f$ for $ Q_{\alpha}({\mathbb R^n})$ spaces
is defined as a Haar wavelet decomposition, with the coefficients
depending on a fixed parameter $\beta>0$. On its support $I_0=[0,
1]^n$, $f(x)$ can be expressed by the binary expansions of the
coordinates of $x$. In particular, $f=f_{\beta}\in Q_{\alpha}({\mathbb
R^n})$ if and only if $\alpha<\beta<\frac{n}{2}$, while for
$\beta=\alpha$, it was shown by Yue and Dafni that $f$ satisfies a
John--Nirenberg inequality for $ Q_{\alpha}({\mathbb R^n})$. When
$\beta\neq 1$, $f$ is a self-affine function. It is continuous almost
everywhere and discontinuous at all dyadic points inside $I_0$. In
addition, it is not monotone along any coordinate direction in any
small cube. When the parameter $\beta\in (0, 1)$, $f$ is onto from
$I_0$ to $[-\frac{1}{1-2^{-\beta}}, \frac{1}{1-2^{-\beta}}]$, and the
graph of $f$ has a non-integer fractal dimension $n+1-\beta$.
Keywords:Haar wavelets, Q spaces, John-Nirenberg inequality, Greedy expansion, self-affine, fractal, Box dimension Categories:42B35, 42C10, 30D50, 28A80 |
8. 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 |
9. CJM 2009 (vol 61 pp. 1182)
Periodic and Almost Periodic Functions on Infinite Sierpinski Gaskets We define periodic functions on infinite blow-ups of the Sierpinski
gasket as lifts of functions defined on certain compact fractafolds
via covering maps. This is analogous to defining periodic functions
on the line as lifts of functions on the circle via covering maps. In
our setting there is only a countable set of covering maps. We
give two different characterizations of periodic functions in terms of
repeating patterns. However, there is no discrete group action that
can be used to characterize periodic functions. We also give a
Fourier series type description in terms of periodic eigenfunctions of
the Laplacian. We define almost periodic functions as uniform limits
of periodic functions.
Category:28A80 |
10. CJM 2009 (vol 61 pp. 1151)
Covering Maps and Periodic Functions on Higher Dimensional Sierpinski Gaskets We construct covering maps from infinite blowups of the
$n$-dimensional Sierpinski gasket $SG_n$ to certain compact
fractafolds based on $SG_n$. These maps are fractal analogs of the
usual covering maps from the line to the circle. The construction
extends work of the second author in the case $n=2$, but a
different method of proof is needed, which amounts to solving a
Sudoku-type puzzle. We can use the covering maps to define the
notion of periodic function on the blowups. We give a
characterization of these periodic functions and describe the
analog of Fourier series expansions. We study covering maps onto
quotient fractalfolds. Finally, we show that such covering maps
fail to exist for many other highly symmetric fractals.
Category:28A80 |
11. CJM 2009 (vol 61 pp. 656)
Generalized Polynomials and Mild Mixing An unsettled conjecture of V. Bergelson and I. H\aa land proposes that
if $(X,\alg,\mu,T)$ is an invertible weak mixing measure preserving
system, where $\mu(X)<\infty$, and if $p_1,p_2,\dots ,p_k$ are
generalized polynomials (functions built out of regular polynomials
via iterated use of the greatest integer or floor function) having the
property that no $p_i$, nor any $p_i-p_j$, $i\neq j$, is constant on a
set of positive density, then for any measurable sets
$A_0,A_1,\dots
,A_k$, there exists a zero-density set $E\subset \z$ such that
\[\lim_{\substack{n\to\infty\\ n\not\in E}} \,\mu(A_0\cap T^{p_1(n)}A_1\cap \cdots
\cap T^{p_k(n)}A_k)=\prod_{i=0}^k \mu(A_i).\] We formulate and prove a
faithful version of this conjecture for mildly mixing systems and
partially characterize, in the degree two case, the set of families
$\{ p_1,p_2, \dots ,p_k\}$ satisfying the hypotheses of this theorem.
Categories:37A25, 28D05 |
12. CJM 2009 (vol 61 pp. 124)
Characterizing Complete Erd\H os Space The space now known as {\em complete Erd\H os
space\/} $\cerdos$ was introduced by Paul Erd\H os in 1940 as the
closed subspace of the Hilbert space $\ell^2$ consisting of all
vectors such that every coordinate is in the convergent sequence
$\{0\}\cup\{1/n:n\in\N\}$. In a solution to a problem posed by Lex G.
Oversteegen we present simple and useful topological
characterizations of $\cerdos$.
As an application we determine the class
of factors of $\cerdos$. In another application we determine
precisely which of the spaces that can be constructed in the Banach
spaces $\ell^p$ according to the `Erd\H os method' are homeomorphic
to $\cerdos$. A novel application states that if $I$ is a
Polishable $F_\sigma$-ideal on $\omega$, then $I$ with the Polish
topology is homeomorphic to either $\Z$, the Cantor set $2^\omega$,
$\Z\times2^\omega$, or $\cerdos$. This last result answers a
question that was asked
by Stevo Todor{\v{c}}evi{\'c}.
Keywords:Complete Erd\H os space, Lelek fan, almost zero-dimensional, nowhere zero-dimensional, Polishable ideals, submeasures on $\omega$, $\R$-trees, line-free groups in Banach spaces Categories:28C10, 46B20, 54F65 |
13. CJM 2008 (vol 60 pp. 1149)
Conjugate Reciprocal Polynomials with All Roots on the Unit Circle We study the geometry, topology and Lebesgue measure of the set of
monic conjugate reciprocal polynomials of fixed degree with all
roots on the unit circle. The set of such polynomials of degree $N$
is naturally associated to a subset of $\R^{N-1}$. We calculate
the volume of this set, prove the set is homeomorphic to the $N-1$
ball and that its isometry group is isomorphic to the dihedral
group of order $2N$.
Categories:11C08, 28A75, 15A52, 54H10, 58D19 |
14. CJM 2008 (vol 60 pp. 457)
Harmonic Coordinates on Fractals with Finitely Ramified Cell Structure We define sets with finitely ramified cell structure, which are
generalizations of post-crit8cally finite self-similar
sets introduced by Kigami and of fractafolds introduced by Strichartz. In general,
we do not assume even local self-similarity, and allow countably many cells
connected at each junction point.
In particular, we consider post-critically infinite fractals.
We prove that if Kigami's resistance form
satisfies certain assumptions, then there exists a weak Riemannian metric
such that the energy can be expressed as the integral of the norm squared
of a weak gradient with respect to an energy measure.
Furthermore, we prove that if such a set can be homeomorphically represented
in harmonic coordinates, then for smooth functions the weak gradient can be
replaced by the usual gradient.
We also prove a simple formula for the energy measure Laplacian in harmonic
coordinates.
Keywords:fractals, self-similarity, energy, resistance, Dirichlet forms, diffusions, quantum graphs, generalized Riemannian metric Categories:28A80, 31C25, 53B99, 58J65, 60J60, 60G18 |
15. 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 |
16. 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 |
17. CJM 2000 (vol 52 pp. 332)
Multiple Mixing and Rank One Group Actions Suppose $G$ is a countable, Abelian group with an element of
infinite order and let ${\cal X}$ be a mixing rank one action of
$G$ on a probability space. Suppose further that the F\o lner
sequence $\{F_n\}$ indexing the towers of ${\cal X}$ satisfies a
``bounded intersection property'': there is a constant $p$ such
that each $\{F_n\}$ can intersect no more than $p$ disjoint
translates of $\{F_n\}$. Then ${\cal X}$ is mixing of all orders.
When $G={\bf Z}$, this extends the results of Kalikow and Ryzhikov
to a large class of ``funny'' rank one transformations. We follow
Ryzhikov's joining technique in our proof: the main theorem follows
from showing that any pairwise independent joining of $k$ copies of
${\cal X}$ is necessarily product measure. This method generalizes
Ryzhikov's technique.
Category:28D15 |
18. 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 |
19. CJM 1998 (vol 50 pp. 1163)
Gradient estimates for harmonic Functions on manifolds with Lipschitz metrics We introduce a distributional Ricci curvature on complete smooth
manifolds with Lipschitz continuous metrics. Under an assumption
on the volume growth of geodesics balls, we obtain a gradient
estimate for weakly harmonic functions if the distributional Ricci
curvature is bounded below.
Categories:60D58, 28D05 |
20. CJM 1998 (vol 50 pp. 638)
Fractals in the large A {\it reverse iterated function system} (r.i.f.s.) is defined to be a
set of expansive maps
$\{T_1,\ldots,T_m\}$ on a discrete metric space $M$. An invariant set
$F$ is defined to be a set satisfying
$F = \bigcup^m_{j=1} T_jF$, and an invariant measure $\mu$ is
defined to be a solution of
$\mu = \sum^m_{j=1} p_j\mu\circ T_j^{-1}$ for positive weights
$p_j$. The structure and basic properties of such invariant sets
and measures is described, and some examples are given.
A {\it blowup} $\cal F$ of a self-similar set $F$ in
$\Bbb R^n$ is defined to be the union of an increasing sequence of
sets, each similar to $F$. We give a general construction of
blowups, and show that under certain hypotheses a blowup is the sum set of
$F$ with an invariant set for a r.i.f.s. Some examples of blowups of
familiar fractals are described. If $\mu$ is an invariant measure
on $\Bbb Z^+$ for a linear r.i.f.s., we describe the behavior of its
{\it analytic} transform, the power series
$\sum^\infty_{n=0} \mu(n)z^n$ on the unit disc.
Category:28A80 |
21. CJM 1997 (vol 49 pp. 1089)
Sets on which measurable functions are determined by their range We study sets on which measurable real-valued functions on a
measurable space with negligibles are determined by their range.
Keywords:measurable function, measurable space with negligibles, continuous image, set of range uniqueness (SRU) Categories:28A20, 28A05, 54C05, 26A30, 03E35, 03E50 |
22. CJM 1997 (vol 49 pp. 3)
Sweeping out properties of operator sequences Let $L_p=L_p(X,\mu)$, $1\leq p\leq\infty$, be the usual Banach
Spaces of real valued functions on a complete non-atomic
probability space. Let $(T_1,\ldots,T_{K})$ be
$L_2$-contractions. Let $0<\varepsilon < \delta\leq1$. Call a
function $f$ a $\delta$-spanning function if $\|f\|_2 = 1$ and if
$\|T_kf-Q_{k-1}T_kf\|_2\geq\delta$ for each $k=1,\ldots,K$, where
$Q_0=0$ and $Q_k$ is the orthogonal projection on the subspace spanned
by $(T_1f,\ldots,T_kf)$. Call a function $h$ a
$(\delta,\varepsilon)$-sweeping function if $\|h\|_\infty\leq1$,
$\|h\|_1<\varepsilon$, and if
$\max_{1\leq k\leq K}|T_kh|>\delta-\varepsilon$ on a set of
measure greater than $1-\varepsilon$. The following is the main
technical result, which is obtained by elementary estimates. There
is an integer $K=K(\varepsilon,\delta)\geq1$ such that if $f$ is a
$\delta$-spanning function, and if the joint distribution
of $(f,T_1f,\ldots,T_Kf)$ is normal, then $h=\bigl((f\wedge
M)\vee(-M)\bigr)/M$
is a $(\delta,\varepsilon)$-sweeping function, for some $M>0$.
Furthermore, if $T_k$s are the averages of operators induced by
the iterates of a measure preserving ergodic transformation, then a
similar result is true without requiring that the joint distribution
is normal. This gives the following theorem on a sequence $(T_i)$ of
these averages. Assume that for each $K\geq1$ there is a subsequence
$(T_{i_1},\ldots,T_{i_K})$ of length $K$, and a $\delta$-spanning
function $f_K$ for this subsequence. Then for each $\varepsilon>0$
there is a function $h$,
$0\leq h\leq1$,
$\|h\|_1<\varepsilon$, such that $\limsup_iT_ih\geq\delta$ a.e..
Another application of the main result gives a refinement of a part
of Bourgain's ``Entropy Theorem'', resulting in a
different, self contained proof of that theorem.
Keywords:Strong and $\delta$-sweeping out, Gaussian distributions, Bourgain's entropy theorem. Categories:28D99, 60F99 |