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