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

arXiv:2611.03778v1 Announce Type: new Abstract: We study quantitative density of orbits of the free semigroup $\Gamma$ generated by two maps of $(0,\infty)$. For the M\"obius 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 $\gamma\in\Gamma$ with \[ |\gamma(x)-u|<C\,\lambda^{-|\gamma|},\qquad \lambda=\exp\!\Big(\frac{\log a\,\log\frac{a}{a-1}}{\log\frac{a^2b}{a-1}}\Big)>1, \] where $|\gamma|$ is the word length of $\gamma$ and $C$ is independent of $\gamma$. 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 $|\gamma(x)-u|<(u+1)(t^{1/|\gamma|}-1)$. 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.

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