Questions tagged [diophantine-approximation]
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125
questions with no upvoted or accepted answers
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Is $\pi$ well-approximable?
Is it known whether, for all $c > 0$, there always exist integers $p$ and $q$ such that
$\left| \pi - \frac{p}{q}\right| < \frac{c}{q^2}$?
This seems like a fundamental question but I couldn't ...
2
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112
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A sequence linked to irrationality
Let $0 < c < 1$ be a real number and $ x \in \mathbb{R}$. We define the sequence $(u_n)_{n \in \mathbb{N}}$ by :
$$u_0 = x$$
$$ \mathrm{If}, u_n \le c, \mathrm{then}, u_{n + 1} = u_n + (1 - c) $$...
2
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112
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The connection of Faltings height and Tate module
Suppose $K$ is a number field, $S$ a finite set of places of $K$, and $A$ is an abelian scheme over $\mathcal O_{K,S}$. I want to ask is there some connections between the Faltings' height $h_F(A)$ ...
2
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131
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A problem raised by Roth's theorem and the notion of approximation exponent
Roth's theorem states that every algebraic irrational has approximation exponent equal to $2$. It follows from Theorem 1 of https://arxiv.org/abs/math/0406300 that the approximation exponent of an ...
2
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134
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"Almost rational" irrational
This is a follow-up to an older question.
Let $r\in \mathbb{R}\setminus\mathbb{Q}$, let $\mathbb{N}$ denote the set of non-negative integers and let $\mathbb{N}_+=\mathbb{N}\setminus\{0\}$. For $n\in\...
2
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0
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86
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On fractional parts and Behrend’s construction
Given $\theta \in \Bbb{T}^D := \Bbb{R}^D/\Bbb{Z}^D$, write $f_\theta$ for the homomorphism from $\Bbb{Z}\to \Bbb{T}^D$ induced by $1\mapsto \theta$.
For $x\in \Bbb{T}^D$, let $||x||$ be the smallest $\...
2
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258
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Baby $abc$ conjecture for $n$-th roots
Is there any progress on a “baby $abc$ conjecture” where you restrict attention to rational approximations of $n$-th roots?
Let $r/s$ be a very close approximation to $(t/u)^{1/n}$, so that
$$
|u\cdot ...
2
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0
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139
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The analogue of Liouville's inequality in several variables
Liouville's Theorem in Diophantine approximation asserts that for each irrational algebraic number $\alpha$ of degree $d$, we have
$$ |q\alpha - p| \gg_\alpha |q|^{1-d}$$
for all integers $p,q \in \...
2
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Is there a way to gain such an estimate?
This problem could be viewed as a polynomial generalization of the Lonely runner conjecture. And $p$, $n$ are taken sufficiently large. Take $n\in \mathbb{N}^*$ fixed, $A_p \subset (\mathbb{Z} / p \...
2
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139
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Double Diophantine approximation
Let $0 < \alpha < 1$. For any $n$ there is a closest lower Diophantine approximation $\max p / q \leq \alpha$ with integer $0 \leq p < q \leq n$. It can be found efficiently, e.g., with Stern-...
2
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88
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Bounds on convergence of two orbits in the limit set of a Schottky group
Suppose we have two points in the limit set of a Schottky group, $x,y\in \Lambda(\Gamma)$. Consider the orbits of those points under a primitive subset $\Gamma'$ of $\Gamma,$ that is, one that does ...
2
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125
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Does $\sum_{i\le k}\mathrm{frac}(n\alpha_i)<1$ hold infinitely often?
For each $t \in \mathbf{R}$, let $\mathrm{frac}(t)$ be its fractional part.
Question. Fix reals $\alpha_1,\ldots,\alpha_k \in (0,1)$ such that $\sum_{i\le k}\alpha_i<1$. Do there exist ...
2
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208
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Examples of essentially sub-linear functions
A dimension function is an increasing, continuous function $%
f:\mathbb R_{+}\rightarrow \mathbb R_{+}$ such that $f(r)\to 0$ as $r\to 0$.
Say that a dimension function $f$ is essentially sub-linear ...
2
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135
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numbers independent over $\mathbb{Q}$ but not BA? numbers that aren't a basis for a number field but are BA?
Has anyone discovered a vector of algebraic real numbers $(a_1,...,a_k)$ such that $1,a_1,...,a_k$ are linearly independent over $\mathbb{Q}$ and such that $(a_1,...,a_k)$ is not "badly approximable"?
...
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The Hausdorff dimensions of variations of Jarnik sets
For $\alpha, \beta>3,$ define
$$\{(x,y)\in[0,1]\times [0,1]: \|qx\|\le q^{1-\alpha}, \|qy\|\le q^{1-\beta} \quad \text{for infinitely many $ q\in \mathbb{N}$}\}.$$
This set can be regarded as a two ...
2
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Discrepancy related independent vector from tensor product?
Here discrepancy is from $(2.4)$ in https://www.ricam.oeaw.ac.at/files/people/siambook_nied.pdf given by 'The discrepancy $D_N(P) = D_N(x_l,\dots,X_N)$ of
the point set $P$ of $N$ points in $\mathbb Z^...
2
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139
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Combination of irrationals
Fix a very small $\epsilon>0$; and irrationals $a_1,a_2>0$. Now suppose we look at all integer combinations of these irrationals which has a small norm; that is,
$$
S=\{(m_1,m_2)\in\mathbb{Z}\...
2
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130
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Quantitative Khintchin's theorem
The quantitative version of Kchintchin's theorem proved by W.Schmidt states that for a.e. $x\in[0,1]$ and every positive integer $K$, if we denote the number of solutions $(p,q)$ to the inequality
$$|...
2
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193
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A high dimensional generation of Dirichlet approximation theorem, linear case and nonlinear case
I am working with something on Diophantine approximation, and I found a high dimensional generation of Dirichlet approximation theorem which may be true; I will be very happy if this is true. The ...
2
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108
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Counting rare events in Diophantine approximation - distributional estimates?
Is there a sort of distributional estimate in Diophantine approximation which allows to estimate the number of solutions which provide a certain quality of approximation? For example, how large is the ...
2
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100
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Zero digits of a p-adic algebraic number
This question might be too simple and I just don't see something very obvious, so I apologize in advance if that is so.
Let $p$ be a prime number and let $\mathbb Z_p$ denote the ring of $p$-adic ...
2
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260
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What are the best current bounds on $\times a \times b$?
Let $a,b \in \mathbb{N}_{\ge 2}$ be two integers that are multiplicatively independent (i.e., are not powers of the same integer). I have seen (Bourgain, Lindenstrauss, Michel, Venkatesh: Some ...
2
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113
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Inhomogeneous approximation mod p
I am interested in the following discrete analog of Kronecker's inhomogeneous approximation theorem.
Given $x_1,\ldots,x_n$ distinct residue classes modulo a prime $p$, and further residue classes $...
2
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106
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Separation of linear forms
Let $r\ge5$ and $0<x_1<x_2<\dots<x_{r-1}<x_r<1$ be independent over $\mathbb Q$. Fix some $N\ge 5$ and consider the linear forms
$$
f({\bf a})=\sum_{i=1}^r a_i x_i,\quad {\bf a}\in \{...
2
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69
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On a sequence of integers
I recall a well-known theorem due to Minkowski.
Theorem. If $\theta$ is irrational and $\alpha$ is not of the form $\alpha = m\theta+n$ for some integers $m$ and $n$, then there are infinitely many ...
2
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247
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Linear forms with best approximation vectors lying in a subspace
Setup: For $u \in \mathbb{R}^n$, let $\rho(u)$ be the Euclidean length, $\sqrt{u_1^2 + \ldots + u_n^2}$. For $x \in \mathbb{R}$ let $\|x\| = \min_{k \in \mathbb{Z}} |x - k|$, and for $x \in \mathbb{R}^...
2
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191
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A question related to metric Diophantine approximation
In metric Diophantine approximation you are often interested in finding conditions on $(\phi(q))_{q \geq 1}$ which guarantee that
$$
\left| \alpha - \frac{p}{q} \right| < \frac{\phi(q)}{q}
$$
has ...
2
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0
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256
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Dalzel's integral for $\pi$ and the lemniscate constant
$\pi=\varpi_2$ can be considered as a member of the sequence of real numbers
$$\varpi_m=2\int\limits_0^1\frac{dt}{\sqrt{1-t^m}},$$
and, for example, the Wallis product formula $$\pi=4\prod_{n=1}^\...
2
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0
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167
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linear forms in abelian logarithms and a conjecture of Lang
Consider the following conjecture, going back to Lang and restated (and proved) in the elliptic case in a 2009 Crelle paper by David and Hirata-Kohno (see Conjecture 1.2 in their paper).
Conjecture (...
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246
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Simultaneous diophantine approximation with polynomial bound
For a given number $\alpha$ continued fractions expansion $(p_n, q_n)$ of $\alpha$ has the remarkable property that not only $|\alpha - \frac{p_n}{q_n}| < \frac{1}{q_n^2}$, but the converse holds - ...
2
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118
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Lower bounds on sums of S-units
Let $S$ be a fixed finite set of valuations on $\mathbb Q$ containing the archimedean one. A $S$-unit is $x\in\mathbb Q$ such that $|x|_v =1$ for all $v\notin S$. For any $S$-units $x_0, \dots, x_n$ ...
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105
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Diophantine approximation away from $0$
Let $\alpha$ be a real irrational algebraic number. The now-classic Thue-Siegel-Roth theorem asserts that for any $\varepsilon > 0$ there exists a positive number $c = c(\alpha, \varepsilon)$ such ...
1
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0
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85
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Motivation for the Weil function for a Cartier divisor
I'm not sure if this is the right place for a question like this.
In Diophantine approximation, on a complex variety $X$ there is a notion of a Weil function for a Cartier divisor $D$ on $X$ which is ...
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284
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Can we avoid all algebraic numbers?
We say a polynomial $p$ in $n$ variables degree at most four, and coefficients $-1,0,1$ is $n$-plain.
We say $x$ is an $n$-plain algebraic number if there exists an $n$-plain polynomial $p$ such that
$...
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0
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87
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Almost modularity of Belyi curves and etale fundamental group of non-Belyi curves
Belyi's theorem states that every curve defined over $\mathbb{\bar Q}$ is almost modular (obtained from $\mathbb{H}^2/\Gamma,\ \Gamma$ a finite index subgroup of $PSL(2,\mathbb{Z})$), after ...
1
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0
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74
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Liouville numbers with some "special" convergents
Recall that a real number $\ell$ is called a Liouville number, if there exists an infinite sequence of rational numbers $(p_n/q_n)_n$ for which
$$
0<\left|\ell-\frac{p_n}{q_n}\right|<\dfrac{1}{...
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81
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Dyson's lemma implies index is small (in proving Roth's theorem)
I am reading the proof of Roth's theorem in Hindry-Silverman's book. In there they used Roth's lemma. I think it is well known that the step of Roth's lemma could be replaced by Dyson's lemma to show ...
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49
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A question on multiplicative diophantine approximation
Suppose $0 <\alpha <1$ is an irrational, and $0 < \gamma_1, \gamma_2 < 1$ are real numbers satisfying $\gamma_i \notin \mathbb{Z} \alpha + \mathbb{Z}$ for $i=1,2$. Consider the sequence $(\...
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0
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166
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Uniform distribution mod $1$ vs independence of random variables
Let $a_1, \cdots, a_k \in [0, 1)$ be real numbers such that $1, a_1, \cdots, a_k$ are independent over the rational numbers. By the Weyl equidistribution criterion in $k$-dimensions, we know that the ...
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0
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138
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A specific Diophantine equation related to the congruent number question
Let $n$ be an odd square free natural number. J.B. Tunnel in his 1983 paper, showed that a number $n$ is congruent, if and only if the number of triples of integers satisfying $2x^2+y^2+8z^2=n$ is ...
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199
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Exponentially well-approximable numbers
Let $\alpha \in [0,1]$ be irrational. I'm interested in the decay rate of $\| n \alpha \|$ as $n \to \infty$, where $\| \cdot \|$ denotes the distance to the nearest integer.
For example:
Dirichlet'...
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0
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73
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On the degree of irrationality of two irrational numbers and their rational (in)dependence
Let $x$ and $y$ be some irrational numbers. If the degree of irrationality of $x$ is the same as that of $y$, is it necessarily the case that $x$ and $y$ are rationally dependent ?
ADDENDUM: What if $...
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139
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On the number of asymptotic solutions of the linear Diophantine equation
Given positive integers $a,b,c$ which are pairwise coprime. Let $f(n)$ denotes the number of the non-negative integer solution $(x,y,z)$ to the equation
$$ax+by+cz=n.$$
we have Prove that there exists ...
1
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0
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137
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Diophantine approximation and the Euclidean algorithm
My question is whether something I've noticed is well-known. It seems like it must be, but I've been unable to find any references that describe what is outlined below.
Given real $x$ and irrational $...
1
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0
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113
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Simultaneous $S$-unit equations
In this question I am primarily interested in rational integers and rational primes, but the same question can be easily extended to number fields.
Let $S = \{p_1, \cdots, p_k\}$ be a finite set of ...
1
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0
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233
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Transcendental functions generating almost integers
Informally speaking, an "almost integer" is a real number very close to an integer.
There are some known ways to construct such examples in a systematic way. One is through the use of certain ...
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0
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95
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Smallest integer lattice point by box measure in a polytope?
Given an integer lattice $\mathcal L\subseteq\mathbb Z^n$ represented by basis $\mathcal B$ and an integer linear program $Ax\leq b$ where $x\in\mathbb Z^n$ is unknown with $A\in\mathbb Z^{m\times n}$ ...
1
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0
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47
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Does a weighted sequence derived from a sequence matching the Siegel lemma BV bound behave as an uniformly random sequence?
Pick integers $r_1',\dots,r_t'$ that achieve the Bombieri Vaaler bound for the Siegel lemma and find an $m$ (it exists by Dirichlet Pigeonhole) such that given prime $T$ gets $m(r_1',\dots,r_t')\equiv(...
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129
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$L^\infty$ norm lower bound for Integer points in null spaces of recursively defined integer vectors?
Letting $\otimes$ be matrix kronecker/tensor product with $n\in\Bbb N$ as a parameter define non-negative integer vectors recursively $$v_1=\begin{bmatrix}a_1&b_1\end{bmatrix}\in\Bbb Z_{>0}^2$$
...
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0
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165
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Analog of Baker's theorem on linear combination of $\log a \log b$
Baker's theorem basically says that, given algebraic numbers $a_1,\ldots,a_n$ and $m_1,\ldots,m_n$, if there is no good reason for a linear combination
$$\sum m_i\log a_i$$
to cancel, then it is ...