A generalization of the Dirichlet approximation theorem for affine actions on the real line
Abstract
We study quantitative density of orbits of the free semigroup Γ generated by two maps of (0,∞). For the Möbius pair x↦ 1/x, x↦ x+1 we show that every orbit is O(n−1)-dense in [0,∞), 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 T0(x)=x/a, T1(x)=bx+1 with a,b>1, which was studied by Bergelson, Misiurewicz and Senti, we prove that for every x>0 and u≥ 0 there are infinitely many γ∈Γ with |γ(x)−u|<C λ−|γ|, λ=exp ((log a loga/(a−1))/(loga2b/(a−1)))>1, where |γ| is the word length of γ and C is independent of γ. This exponential rate holds whether or not loga 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)(t1/|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.
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BibTeX
@misc{javaheri2026generalization,
title = {{A generalization of the Dirichlet approximation theorem for affine actions on the real line}},
author = {Mohammad Javaheri},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03778},
eprint = {2611.03778},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03778},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A generalization of the Dirichlet approximation theorem for affine actions on the real line
AU - Mohammad Javaheri
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03778
UR - https://arxiv.org/abs/2611.03778
AB - We study quantitative density of orbits of the free semigroup Γ generated by two maps of (0,∞). For the Möbius pair x↦ 1/x, x↦ x+1 we show that every orbit is O(n^−1)-dense in [0,∞), 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≥ 0 there are infinitely many γ∈Γ with |γ(x)−u|<C λ^−|γ|, λ=exp ((log a loga/(a−1))/(loga^2b/(a−1)))>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.
ER -