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On a sumset conjecture of Erdős

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

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