REVIEW
A generalization of the Dirichlet approximation theorem for affine actions on the real line
Number Theory · Submitted 2 Nov 2026 · 18 references
Pith's verdict: Solid quantitative density results for two semigroup actions; main theorem (exponential rate for affine pair) is a real step forward, and the proofs are clean.
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Referee report
The paper
The paper claims
Every orbit of a free semigroup generated by two affine maps is exponentially dense, and for the Möbius pair the rate of approximation depends on continued fractions.
The reading
What carries the argument
For the Möbius pair, the machinery is the continued fraction expansion of rational numbers and the Stern–Brocot tree, which give explicit word constructions and a lower bound via the sum of partial quotients. For the affine pair, the key is a normal form for γ(x) as a sum of powers (Lemma 3.1) and a greedy algorithm that selects exponents ℓ_j to build a word whose approximation error decays exponentially; the exponent λ emerges from balancing the parameters a and b.
Extended reading notes
Core claim
The central claim is that the orbit of any positive real number under the free semigroup generated by the affine maps T_0(x)=x/a and T_1(x)=bx+1 (a,b>1) is exponentially dense: for every x>0 and u≥0 there exist infinitely many words γ with |γ(x)−u| < C $λ^{{−|γ|}}$, where λ = exp( (log a · log(a/(a-1))) / (log($a^{2}$ b/(a-1))) ) > 1 and C depends on a,b,x,u but not on γ. This improves the previously known denseness result of Bergelson, Misiurewicz and Senti from a qualitative to a quantitative exponential rate. For the Möbius pair, the paper establishes a complete quantitative picture: O($n^{{−1}}$) denseness for all targets, a rate $n^{{−2}}$ for irrational targets, exponential rates for targets with bounded partial quotients, and optimality of the $n^{{−1}}$ rate for rational targets from rational starting points.
Load-bearing premise
For the Möbius pair, the optimality of the $n^{{−1}}$ rate for rational targets is proved only when the starting point is rational; the authors state they expect the same bound for irrational starting points but have not proved it.
Editorial extensions
If this is right
- For the affine pair with a=2, b=3, the result implies that the Collatz orbit starting from any positive real number can be made exponentially close to 1; this does not resolve the Collatz conjecture but shows the semigroup orbit is extremely dense.
- For the Möbius pair, the rate at which a given irrational target is approached is completely determined by its continued fraction expansion, and for almost every target the rate is exp(−c n / log n), which is optimal up to the constant c.
- The corrected Kronecker-type estimate (Theorem 3.2) gives an explicit O(n^{−1}) bound with a simple construction when log_a b is irrational, though it is superseded by the exponential bound.
- The prime approximation discussion suggests that for every u>40/19 there is an infinite chain of primes whose logarithms approximate u in a strong sense, and that under the Riemann Hypothesis this holds for every u>2.
Reading between the lines
- The exponential density result for the affine pair may have applications in diophantine approximation on fractals or in constructing points with prescribed rates of approximation in one-dimensional dynamics.
- The optimality for rational targets in the Möbius pair is proved only for rational starting points; extending this to irrational starting points would complete the picture, as the authors note is expected.
- The paper’s prime approximation section is more a survey of known results than a new theorem; the connection to Dirichlet approximation via semigroups is not directly used there.
- The constant λ in Theorem 3.8 is generally small (e.g., λ(2,3)≈1.213), so the exponential density is modest but still far better than polynomial density.
Original 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.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study quantitative density of orbits of the free semigroup $\Gamma$ generated by two maps of $(0,\infty)$. | contribution 0.88 | 0.86 | 0.04 | asserted 0.73 |
| s1 | 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. | result 0.99 | 0.95 | 0.23 | asserted 1.00 |
| s2 | 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 $\gamma$. | result 0.97 | 0.95 | 0.42 | asserted 1.00 |
| s3 | This exponential rate holds whether or not $\log_a b$ is rational, and the exponential order cannot be improved for almost every $u$. | result 0.95 | 0.90 | 0.17 | asserted 1.00 |
| s4 | We also give a corrected version of the Kronecker-type estimate $|\gamma(x)-u|<(u+1)(t^{1/|\gamma|}-1)$. | result 0.57 | 0.91 | 0.05 | asserted 1.00 |
| s5 | 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. | contribution 0.95 | 0.72 | 0.04 | asserted 0.77 |
What was examined, and what was not
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References
Discussion
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