Abstract view
On a Sumset Conjecture of Erdős


Published:20140609
Printed: Aug 2015
Mauro Di Nasso,
Dipartimento di Matematica, Universita' di Pisa, Largo Bruno Pontecorvo 5, Pisa 56127, Italy
Isaac Goldbring,
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Science and Engineering Offices M/C 249, 851 S. Morgan St., Chicago, IL 606077045, USA
Renling Jin,
Department of Mathematics, College of Charleston, Charleston, SC 29424, USA
Steven Leth,
School of Mathematical Sciences, University of Northern Colorado, Campus Box 122, 510 20th Street, Greeley, CO 80639, USA
Martino Lupini,
Department of Mathematics and Statistics, York University, N520 Ross, 4700 Keele Street, M3J 1P3, Toronto, ON
Karl Mahlburg,
Department of Mathematics, Louisiana State University, 228 Lockett Hall, Baton Rouge, LA 70803, USA
Abstract
Erdős 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 Erdős' 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 Erdős'
conjecture for subsets of the natural numbers that are pseudorandom.
MSC Classifications: 
11B05, 11B13, 11P70, 28D15, 37A45 show english descriptions
Density, gaps, topology Additive bases, including sumsets [See also 05B10] Inverse problems of additive number theory, including sumsets General groups of measurepreserving transformations Relations with number theory and harmonic analysis [See also 11Kxx]
11B05  Density, gaps, topology 11B13  Additive bases, including sumsets [See also 05B10] 11P70  Inverse problems of additive number theory, including sumsets 28D15  General groups of measurepreserving transformations 37A45  Relations with number theory and harmonic analysis [See also 11Kxx]
