A generalization of the Dirichlet approximation theorem for affine actions on the real line

We study quantitative density of orbits of the free semigroup $Γ$ generated by two maps of $(0,\infty)$. For the Möbius pair $x\mapsto 1/x$, $x\mapsto x+1$ we show that every orbit is $O(n^{-1})$-dense in $[0,\infty)$, that every irrational number $u$ is approached at a rate governed by the continued fraction of $u$ (at least $n^{-2}$, exponential when $u$ has bounded partial quotients, and $\exp(-cn/\log n)$ for almost every $u$), and that for rational $u$ and rational starting points the rate $n^{-1}$ is optimal. For the affine pair $T_0(x)=x/a$, $T_1(x)=bx+1$ with $a,b>1$, which was studied by Bergelson, Misiurewicz and Senti, we prove that for every $x>0$ and $u\ge 0$ there are infinitely many $γ\inΓ$ with $|γ(x)-u|<C\,λ^{-|γ|},\qquad λ=\exp\!\Big(\frac{\log a\,\log\frac{a}{a-1}}{\log\frac{a^2b}{a-1}}\Big)>1,$ where $|γ|$ is the word length of $γ$ and $C$ is independent of $γ$. This exponential rate holds whether or not $\log_a b$ is rational, and the exponential order cannot be improved for almost every $u$. We also give a corrected version of the Kronecker-type estimate $|γ(x)-u|<(u+1)(t^{1/|γ|}-1)$. Finally, we discuss the approximation of real numbers by ratios of primes and by ratios of logarithms of primes, updated in light of recent results on gaps between primes.