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On a Sumset Conjecture of Erdős

Published online by Cambridge University Press:  20 November 2018

Mauro Di Nasso ,
Isaac Goldbring ,
Renling Jin ,
Steven Leth ,
Martino Lupini  and
Karl Mahlburg
Mauro Di Nasso
Affiliation:
Dipartimento di Matematica, Universita' di Pisa, Largo Bruno Pontecorvo 5, Pisa 56127, Italy e-mail: dinasso@dm.unipi.it
Isaac Goldbring
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL 60607-7045, USA e-mail: isaac@math.uic.edu
Renling Jin
Affiliation:
Department of Mathematics, College of Charleston, Charleston, SC 29424, USA e-mail: JinR@cofc.edu
Steven Leth
Affiliation:
School of Mathematical Sciences, University of Northern Colorado, Greeley, CO 80639, USA e-mail: Steven.Leth@unco.edu
Martino Lupini
Affiliation:
Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3 e-mail: mlupini@mathstat.yorku.ca
Karl Mahlburg
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA e-mail: mahlburg@math.lsu.edu
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Abstract

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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 where $A$ 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

11B0511B1311P7028D1537A45sumsets of integersasymptotic densityamenable groupsnonstandard analysis
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 4 , 01 August 2015 , pp. 795 - 809
Copyright
Copyright © Canadian Mathematical Society 2015

References

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