Contractions and expansion
Combinatorics
2011-12-16 v1 Number Theory
Abstract
Let A be a finite set of reals and let K >= 1 be a real number. Suppose that for each a in A we are given an injective map f_a : A -> R which fixes a and contracts other points towards it in the sense that |a - f_a(x)| <= |a - x|/K for all x in A, and such that f_a(x) always lies between a and x. Then the union of the f_a(A) has cardinality >= K|A|/10 - O_K(1). An immediate consequence of this is the estimate |A + K.A| >= K|A|/10 - O_K(1), which is a slightly weakened version of a result of Bukh.
Cite
@article{arxiv.1112.3468,
title = {Contractions and expansion},
author = {Emmanuel Breuillard and Ben Green},
journal= {arXiv preprint arXiv:1112.3468},
year = {2011}
}
Comments
6 pages, submitted to special volume of EJC in honour of Yahya Hamidoune