8 research outputs found
On a sumset conjecture of Erd\H{o}s
Erd\H{o}s conjectured that for any set with positive
lower asymptotic density, there are infinite sets
such that . We verify Erd\H{o}s' conjecture in the case that
has Banach density exceeding . As a consequence, we prove
that, for with positive Banach density (a much weaker
assumption than positive lower density), we can find infinite such that is contained in the union of and a translate of
. Both of the aforementioned results are generalized to arbitrary countable
amenable groups. We also provide a positive solution to Erd\H{o}s' conjecture
for subsets of the natural numbers that are pseudorandom.Comment: 17 pages; new version has a slightly different title, some minor
typos are fixed, and the exposition of Lemma 4.6 has been improved. To appear
in the Canadian Journal of Mathematic
Problems on infinite sumset configurations in the integers and beyond
In contrast to finite arithmetic configurations, relatively little is known
about which infinite patterns can be found in every set of natural numbers with
positive density. Building on recent advances showing infinite sumsets can be
found, we explore numerous open problems and obstructions to finding other
infinite configurations in every set of natural numbers with positive density.Comment: 37 page
A proof of a sumset conjecture of Erd\H{o}s
In this paper we show that every set with positive
density contains for some pair of infinite subsets of ,
settling a conjecture of Erd\H{o}s. The proof features two different
decompositions of an arbitrary bounded sequence into a structured component and
a pseudo-random component. Our methods are quite general, allowing us to prove
a version of this conjecture for countable amenable groups.Comment: 54 pages. Corrected proof of Theorem 3.22 and added Example 3.27
Keywords: sum sets, almost periodic functions, ultrafilter
Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory
The goal of this present manuscript is to introduce the reader to the
nonstandard method and to provide an overview of its most prominent
applications in Ramsey theory and combinatorial number theory.Comment: 126 pages. Comments welcom