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# On a Sumset Conjecture of Erdős

Published:2014-06-09
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 60607-7045, 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
 Format: LaTeX MathJax PDF

## 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.
 Keywords: sumsets of integers, asymptotic density, amenable groups, nonstandard analysis
 MSC Classifications: 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 measure-preserving transformations 37A45 - Relations with number theory and harmonic analysis [See also 11Kxx]

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