# Questions tagged [diophantine-approximation]

The diophantine-approximation tag has no usage guidance.

The diophantine-approximation tag has no usage guidance.

307
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Let $b_1,\cdots,b_s$ be not all zero complex numbers, $a_1,\cdots,a_s$ be distinct numbers.
Can one assert there exists a constant $c>0$ such that for any positive integer $n$
$$\left \lvert \sum_{...

1
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0
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67
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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 ...

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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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Consider the linear diophantine equation
$$\alpha u+\beta v =r+ \delta$$
where $\alpha,\beta,r\in\mathbb Q$ are known and their binary expansion has $O(k)$ bits to exactly represent them and $\delta\...

-2
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1
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128
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A matrix $$\begin{bmatrix}w &x \\\ y &z\end{bmatrix}\in\mathbb Z^{2\times 2}$$ is unimodular if $$|wz-xy|=1$$ holds.
Is there a parametrization of such matrices with $2wy>(wz+xy)$ and $2xz&...

2
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1
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129
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We have an algebraic number $a$ and a real number $b$. Can the following inequality have infinitely many solutions for $n \in \mathbb{N}$?
$$ \{an\} \in [b - \frac{1}{2^n}, b + \frac{1}{2^n}] $$
Here $...

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0
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50
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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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73
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MO-post
Why Is $\frac{163}{\operatorname{ln}(163)}$ a Near-Integer?
presents the topic in a somewhat chaotic manner. However, this topic should admit a regular diophantine analysis of the ...

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1
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134
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Let $0 \neq \alpha \in [0,1]$ and $q$ a positive integer.
Let $||.||$ denote the distance to the closest integer and define
$$
N_i(q) = \sum_{ \substack{ -q/2 \leq n \leq q/2 \\ \frac{i}{q} \leq || \...

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49
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There are known methods for finding integer solutions to Thue equations, that is, equations of the form $f(x,y)=0$, where $f$ is an irreducible homogeneous polynomial of degree $d\geq 3$. Are there ...

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410
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We know that the time spent by the sequence $na \mod 1$, $n$ ranging from $1$ up to $x$ and $a$ irrational, at any interval of length $\delta$ is approximately $\delta x$. There are known results when ...

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210
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1. Question
How to get from the formulas
$$ \left| \frac{\log 2}{\log 3} - \frac{p}{q} \right| < c_a\frac{1}{q^{B_a}} \ \ \ \ \ \ \ \ \ \ \ (1.1)$$
and / or
$$ \left| \frac{\log 2}{\log 3} - \frac{...

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1
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79
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Let $r\in [0,1]\setminus\mathbb{Q}$, let $\mathbb{N}$ denote the set of non-negative integers and let $\mathbb{N}_+=\mathbb{N}\setminus\{0\}$. For $n\in\mathbb{N}_+$ let $$\alpha_r(n)=\min\{\big|r-\...

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25
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Producing small solutions to underdetermined, inhomogenous integer linear systems is a hard problem with wide applications in diophantine approximation. A simpler subproblem is the following.
Let $\...

3
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1
answer

199
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Given any $\epsilon > 0$, are there infinitely many $(a,b) \in \mathbb{Z}^2$ with $(a,b) = 1$ such that $$\left|\pi - \frac{a}{b}\right| < \frac{\epsilon}{b^2}?$$
According to this document, if ...

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44
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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 $(\...

3
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53
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Let $\mathcal L$ be the space of unimodular (covolume one) lattices in $\mathbb R^d$. The $i$-th successive minimum of $L\in \mathcal L$, denoted $\lambda_i(L)$ is the infimum of the radii of the ...

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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 ...

2
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2
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Let $\mathcal L$ be the space of unimodular (covolume one) lattices in $\mathbb R^d$. The $i$-th successive minimum of $L\in \mathcal L$, denoted $\lambda_i(L)$ is the infimum of the radii of the ...

9
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512
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This question is inspired by xkcd #2585 (Rounding):
Let $u_0,\ldots,u_n$ be positive real numbers (we can assume w.l.o.g. that $u_0=1$) or “units”.
Consider the following directed graph: its vertices ...

2
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1
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178
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I am looking for an upper bound $R=R_{n,\varepsilon}$ such that for given $\varepsilon>0$ and real numbers $\alpha_1, \dotsc, \alpha_n$ in, say, $[1,2]$, there is $x\in [1,R]$ such that
$$
\frac 1n\...

6
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1
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315
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A badly approximable number is an $x$ for which there is a positive constant $c$ such that for all rational $p/q$ we have
$$\left|{ x - \frac{p}{q} }\right| > \frac{c}{q^2} \ . $$
The set of badly ...

2
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0
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356
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Are there any solutions in positive integers of
$x^3 + 1 = (x - k) y^3$?
The closest I can get is
$19^3 + 1 = 20 \times 7^3$,
but $20\gt 19$ so it just misses!
For the related
$x^3 - 1 = (x - k) y^3$,...

2
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0
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230
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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 ...

3
votes

1
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224
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Let $\alpha \in \mathbb{R}$ and $N$ a positive integer. I am interested in the quantity
$$
D(\alpha, N) := \# \{ n \in [1, N]: \| n \alpha \| < 1/N \},
$$
$\| x \|$ denotes the distance to the ...

3
votes

2
answers

104
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Let $G=\text{SL}(d,\mathbb R)$ and $\Gamma = \text{SL}(d,\mathbb Z)$. The homogeneous space is identified with the space of unimodular lattices, denoted $X_d$. Let $Y_d$ denote the space of unimodular ...

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290
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Here $a,b,c,d,e$ are distinct and all greater than $1$.
This question was formerly posted on Math.Stackexchange, precisely here,
but seems to be more general than some other tough number theory ...

2
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1
answer

173
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In the book Neverending fractions from Borwein, van der Poorten, Shallit and Zudilin, there is the so called distance formula (Theorem 2.45, p. 43) stated:
$$\alpha_1\alpha_2\cdot...\cdot\alpha_n=\...

4
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1
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396
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Intuitively, a constructively irrational number is one for which we can effectively separate it from any rational number in terms of the latter's denominator. More formally, a constructively ...

2
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1
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261
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Let $\| \cdot \|$ denote the maximum norm in Euclidean spaces.
Consider the set $D_{m,n}$ of $m \times n $ real matrices satisfying that the system of inequalities
$$\|Aq-p\|^m < \frac{1}{T}, \|q\|^...

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$\newcommand{\IR}{\mathbb R}$
$\newcommand{\IT}{\mathbb T}$
$\newcommand{\w}{\omega}$
$\newcommand{\e}{\varepsilon}$
Taras Banakh and me proceed a long quest answering a question of ougao at ...

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135
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This is inspired by a recent math.SE question.
Given that mathematicians like to come up with theoretical constructs which do not necessarily always have any practical purpose (but sometimes provide ...

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1
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182
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EDIT (August 9, 2021): I would like to ask a more general question. The original question that was fully answered is below the line.
For a positive real number $x$, denote the fractional part $x-[x]$ ...

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319
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Let $m,n$ be natural numbers such that $2^n - 1 \mid 3^m - 1$. By results from Bugeaud-Corvaja-Zannier, say Theorem 3 of this paper , we know that for any constant $C > 0$ we must have $m > Cn$ ...

2
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129
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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 \...

3
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1
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142
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Do there exist an algorithm, which, given a polynomial $P(x,y)$ with integer coefficients, determines whether Diophantine equation $P(x,y)=0$ has finite or infinite number of integer solutions?
Famous ...

2
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0
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100
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The well known result of Hurwitz on Diophantine approximation says that for any irrational $\alpha$ there are infinitely many integer numbers $p$ and $q$ such that
$$
|\alpha -\frac{p}{q}|<\frac{1}{...

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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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122
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I am a first year graduate student and I am eager to work on irrationality/transcendental proofs of specific numbers like Euler's constant gamma. Because backgrounds for Elliptic Curves include very ...

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130
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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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Let $\vec x = (x_1,x_2,\dots, x_k) \in \mathbb{R}^k$.
Dirichlet’s theorem guarantees that for each $N$, there exists $(n_0,n_1,n_2,\dots,n_k) \in \mathbb{Z}^{k+1} \setminus \{\vec 0\}$ with $\max(|n_1|...

3
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138
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A famous conjecture of Littlewood asserts that for each $\alpha, \beta \in \mathbb{R}$ and $\varepsilon > 0$, the set
$$ A(\alpha,\beta;\varepsilon) = \{ n \in \mathbb{N} \ : \ \Vert \alpha n \Vert ...

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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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1
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147
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If $q = [q_0;q_1 \dots]$, say $q_i$ is the $i$-th partial quotient of $q$. My question is the following:
Can one construct an explicit example of irrational $r,s > 0$ such that
$\{ 1,r,s\}$ is $\...

4
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1
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193
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For a real number $x$, we denote
$$ \|x\|=\inf_{m\in {\Bbb Z}}|x+m|.$$
Problem 1:
Roth's theorem states that given any irrational algebraic number $\alpha$ and for any $\epsilon>0$, there exists a ...

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0
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135
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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 ...

2
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1
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302
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Let $$u_n=\prod_{k=1}^{n-1}\cos^2(k)$$ then $$\frac1n \ln(u_n) = \frac1n\sum_{k=0}^{n-1} \ln(\cos^2(k)) \underset{n\to\infty}\longrightarrow \frac1{2\pi} \int_0^{2\pi} \ln(\cos^2(x))\,{\rm d}x = -\ln(...

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2
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248
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I am trying to read this letter by Sarnak, where, among other things, he discusses the problem of intrinsic diophantine approximation on $S^3 = \left\{\mathbf{x} \in \mathbb{R}^4 : x_1^2+x_2^2+x_3^2+...

4
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1
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194
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Let $G:=SL(m+n,\mathbb R)$ and $\Gamma :=SL(m+n,\mathbb Z)$ and $X:=G/\Gamma$.
(1) Let $M$ denote the set of all $m \times n$ matrices with real entries. A matrix $A \in M$ is called $\textit{singular}...

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I am looking for an irrational algebraic number $\alpha \in [0,1[$ whose base two expansion
$$
\alpha = \sum_{i=1}^\infty {1 \over 2^{\varphi(i)}},
$$
is easily computable. By this I mean $\varphi : {...