Contractions and expansion

TL;DR

This paper investigates the properties of contractions and expansions in finite sets of real numbers, establishing bounds on the size of unions of contracted sets and related sumset estimates.

Contribution

It introduces a new framework for analyzing contractions towards points in finite sets and derives a lower bound on the union of contracted images, extending previous sumset results.

Findings

Union of contracted images has size at least K|A|/10 minus a constant.

Derived a lower bound for |A + K·A| similar to Bukh's result.

Established conditions for injective contractions fixing points in finite sets.

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.