A convex set with a rich difference

Oliver Roche-Newton; Audie Warren

A convex set with a rich difference

Combinatorics 2022-08-19 v2

Authors: Oliver Roche-Newton , Audie Warren

Abstract

We construct a convex set $A$ with cardinality $2n$ and with the property that an element of the difference set $A-A$ can be represented in $n$ different ways. We also show that this construction is optimal by proving that for any convex set $A$ , the maximum possible number of representations an element of $A-A$ can have is $\lfloor |A|/2 \rfloor$ .

Keywords

set theory and dimension theory

Cite

@article{arxiv.2208.03258,
  title  = {A convex set with a rich difference},
  author = {Oliver Roche-Newton and Audie Warren},
  journal= {arXiv preprint arXiv:2208.03258},
  year   = {2022}
}

Comments

This version has been edited to include a reference to a paper of Schoen, which we were not previously aware of. The paper of Schoen proves a similar result, with a worse multiplicative constant

A convex set with a rich difference

Abstract

Keywords

Cite

Comments

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