On orbit sets generated by semigroups of one-dimensional affine functions

TL;DR

This paper investigates the growth and density of orbit sets generated by semigroups of one-dimensional affine functions, establishing bounds and conditions under which these sets have positive density.

Contribution

It provides new lower bounds on the size of orbit sets and characterizes when these sets have positive density, extending previous upper bound results.

Findings

Established a lower bound (x^{}) for multiset sizes of orbit sets.

Proved that orbit sets can have positive density under certain algebraic conditions.

Extended previous bounds to cases involving exact covering systems of integers.

Abstract

The one-dimensional orbit set $\langle F : s \rangle$ is formed by the images of a number $s$ under the action of a semigroup generated by integer affine functions $f_i=a_i x+b_i$ taken from the set $F=\{f_1,\ldots,f_n\}$. P.Erd\H{o}s established an upper bound $O(x^{\sigma+\epsilon})$ for the growth function $|\langle F : s \rangle\cap[0,x]|$, where $1/a_1^{\sigma}+1/a_2^{\sigma}+\ldots + 1/a_n^{\sigma}=1$ and $\varepsilon>0$, which was extended to orbit multisets and real affine functions by J.Lagarias. We complement this by a lower bound $\Omega(x^{\sigma})$ for the multiset size $|\langle F : s \rangle^\#\cap[0,x]|$. P.Erd\H{o}s and R.Graham asked whether an orbit set $\langle F : s \rangle$ has positive density when $F$ is a basis of a free semigroup and $1/a_1+1/a_2+\ldots + 1/a_n=1$. Under these two conditions, we establish a sublinear lower bound $|\langle F : s \rangle \cap [0,x]|=\Omega(x/\log^{\frac{n-1}2} x)$. We also show that in the case when the functions of $F$ form an exact covering system of integers, i.e. when $f_1(\mathbb Z) \sqcup \ldots \sqcup f_n(\mathbb Z)=\mathbb Z$, this bound can be strengthened to $\Omega(x)$, so the set $\langle F : s \rangle$ has positive density.