On the Structure of Sets of Large Doubling

Allison Lewko; Mark Lewko

On the Structure of Sets of Large Doubling

Classical Analysis and ODEs 2011-03-01 v4 Combinatorics

Authors: Allison Lewko , Mark Lewko

Abstract

We investigate the structure of finite sets $A \subseteq \Z$ where $|A+A|$ is large. We present a combinatorial construction that serves as a counterexample to natural conjectures in the pursuit of an "anti-Freiman" theory in additive combinatorics. In particular, we answer a question along these lines posed by O'Bryant. Our construction also answers several questions about the nature of finite unions of $B_2[g]$ and $B^\circ_2[g]$ sets, and enables us to construct a $\Lambda(4)$ set which does not contain large $B_2[g]$ or $B^\circ_2[g]$ sets.

Keywords

additive combinatorics large doubling Freiman theory B2[g] sets Lambda(4) sets finite unions combinatorial construction

Cite

@article{arxiv.1003.4561,
  title  = {On the Structure of Sets of Large Doubling},
  author = {Allison Lewko and Mark Lewko},
  journal= {arXiv preprint arXiv:1003.4561},
  year   = {2011}
}

Comments

23 pages, changed title, revised version reflects work of Meyer that we were previously unaware of

On the Structure of Sets of Large Doubling

Abstract

Keywords

Cite

Comments

Related papers