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

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.