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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 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 |