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

TL;DR

This work analyzes one-dimensional orbit sets generated by a finite family of affine maps f_i(x)=a_i x+b_i with a_i≥1, b_i≥0. Building on Erdős–Lagarias upper bounds for orbit multisets, the authors derive a matching-type lower bound for the multiset size |⟨F:S⟩#∩[0,x]| and establish a sublinear lower bound for the density of orbit sets under the Erdős–Graham framework when the slopes form a free semigroup with ∑ 1/a_i = 1. They further show that if the maps form an exact covering system, i.e., f_i(Z) partition Z, then the orbit set has positive density, with a linear lower bound; their approach combines combinatorial constructions of equal-slope compositions with Stirling-type estimates, and a geometric Ping-Pong framework via Klarner tuples to certify freeness and handle exact coverings. Together, these results address Erdős–Graham questions and illuminate the density behavior of orbit sets under various structural conditions.

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.Erdős established an upper bound $O(x^{σ+ε})$ for the growth function $|\langle F : s \rangle\cap[0,x]|$, where $1/a_1^σ+1/a_2^σ+\ldots + 1/a_n^σ=1$ and $\varepsilon>0$, which was extended to orbit multisets and real affine functions by J.Lagarias. We complement this by a lower bound $Ω(x^σ)$ for the multiset size $|\langle F : s \rangle^\#\cap[0,x]|$. P.Erdős 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 $|\langle F : s \rangle \cap [0,x]|=Ω(x/\log^{\frac{n-1}2} x)$. 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 $Ω(x)$, so the set $\langle F : s \rangle$ has positive density.

Theorems & Definitions (12)

• Theorem 1: Erdős--Lagarias Upper Bound
• Theorem 2
• Remark 3
• Lemma 4
• proof
• proof : Proof of Theorem \ref{['lowerbound']}.
• Theorem 5
• Lemma 6
• proof
• proof : Proof of Theorem \ref{['subbound']}