Questions tagged [diophantine-approximation]

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3
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146 views

Does the equation $a^b+b^c+c^a=d^e$ have solutions in $\mathbb {N}$

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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1answer
120 views

Distance formula for continued fractions

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=\...
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1answer
307 views

What is known about constructively irrational numbers?

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 ...
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1answer
110 views

Almost every $m\times n$ real matrix is Dirichlet approximable

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\|^...
6
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1answer
219 views

A property of rapid sequences of natural numbers

$\newcommand{\IR}{\mathbb R}$ $\newcommand{\IT}{\mathbb T}$ $\newcommand{\w}{\omega}$ $\newcommand{\e}{\varepsilon}$ Taras Banakh and me proceed a long quest answering a question of ougao at ...
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1answer
132 views

A pathological (?) function involving powers

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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1answer
170 views

Bounding the fractional parts of the $p^{\text{th}}$ roots of $n,n^2,...,n^{p-1}$

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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301 views

Given $2^n - 1 \mid 3^m - 1$, how large must $m$ be compared to $n$?

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 $m > Cn$ ...
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114 views

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 \...
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1answer
107 views

Determine if a 2-variable Diophantine equation has a finite or infinite number of solutions

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 ...
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65 views

What is the sequence of badly approximable numbers to omit in Hurwitz' second Theorem for Diophantine Approximation to obtain better constants?

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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87 views

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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114 views

What are some interesting problems in the intersection of Diophantine Approx and Algebraic Geometry?

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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73 views

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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88 views

Optimal exponent in Dirichlet’s theorem on diophantine approximation

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|...
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135 views

A randomised variant of the Littlewood conjecture

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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59 views

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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1answer
113 views

Bounded, aperiodic irrationals with bounded, aperiodic sum

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 $\...
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1answer
161 views

More about Roth's theorem: bound for the constant and multidimensional case

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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129 views

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 ...
2
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1answer
294 views

Find better than $ 4^n\prod_{k=1}^{n-1}\cos^2(k)\sim e^{o(n)}$

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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2answers
232 views

Diophantine approximation on spheres

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+...
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1answer
120 views

The Hausdorff dimension of $F^+_{m,n}$ singular points

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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89 views

algebraic number with explicit base two digits

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 : {...
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112 views

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 $...
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60 views

Discrepancy estimate for $3$-interval exchange or $n$-interval exchange map, $n\geq 3$

We know that 2-interval exchange on $\mathbb{T}$($\mathbb{T}$ is identified with $[0,1]$ for convenient in the follow context) is just a rotation on $\mathbb{T}$, and there is a process called ...
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131 views

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 \...
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88 views

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 ...
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84 views

Explicit constructions of $\tau$-approximable numbers

A number $x\in\mathbb R$ is said to be $\tau$-approximable if there are $c,C>0$ such that for infinitely many couples of integers $(p,q)$, $$|x-\frac pq|<Cq^{-\tau}$$ and for all couples $(p,q)$ ...
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69 views

Distribution of the values of the product $\prod_{k=1}^n |1-e(k\alpha)|$ for an irrational number $\alpha$

For an irrational number $\alpha$, let $e(k\alpha):=\exp(2k\pi i\alpha)$. It was indicated in this thread that $$\limsup_{n \to \infty} \prod_{k=1}^n |1-e(k\alpha)|=\infty$$ (actually a weaker result ...
5
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1answer
200 views

Irrationality measure of powers

Let $\alpha$ be an irrational number. Denote by $\mu(\alpha)$ its irrationality measure. Can one say anything about $\mu(\alpha^n)$ for every $n\in\mathbb N$? Even more, one knows that $\mu(e)=2$. Can ...
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37 views

magnitude of the best approximants denominator for an irrational number

If the irrationality measure of an irrational number $x\in \mathbb{R}$ is some $\mu>0$, it implies that for $\mu'<\mu$ there are infinitely many couples of integers $(p_n,q_n),n\in \mathbb{N}$ ...
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94 views

Kronecker's theorem on diophantine approximation for $\mathrm{SL}_2(\mathbb{Z})$

Kronecker's theorem on diophantine approximation gives a criterium whether the orbit of a tupple ${\underline\alpha}=(\alpha_1,\ldots,\alpha_n)\in \mathbb{R}^n/\mathbb{Z}^n$ comes arbitrarily close to ...
3
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1answer
289 views

Close integer solutions to $ab-cd=1$?

I am looking for infinite set of Diophantine solutions. Suppose we require $$0<\min(a,d)<\max(a,d)<\min(b,c)<\max(b,c)\leq\sqrt 2\min(a,d)$$ $$a,b,c,d\in\mathbb Z$$ then can we still find ...
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1answer
194 views

References for irrational random walks

I am interested in the symmetric random walk on $\mathbb{R}$ which increments have the discrete law $$\mu=\sum_{i=1}^q p_i (\delta_{\omega_i}+\delta_{-\omega_i})$$ where the $p_i$ sum to $1/2$ and the ...
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0answers
73 views

Working with truncation of inverse of integers (number of necessary digits)

Ideally I would like to find exact value of $Mr'$ where $r'=\frac{1}{M'}$ where $M'$ ranges from $1$ to $T+1$ and $1\leq M\leq T$ holds. However in real world approximations have finite precision. If ...
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2answers
255 views

Chebyshev rational approximation of $e^{x}, x >0$: does it exist?

It's well known that the scalar function $e^x$, for $x \in (-\infty,0]$ can be approximated by Chebyshev rational approximation. In practice, one wants to use a partial fraction decomposition form ...
5
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1answer
568 views

Arithmetic-geometric mean for rationals?

Let $\operatorname{AGM}(x,y)$ be the arithmetic-geometric mean of $x$ and $y$. Given an error $\varepsilon>0$, a bound $b\in\mathbb R_+$ and a function $f:\mathbb R\rightarrow\mathbb R$ with $f(x)=...
3
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1answer
267 views

On weaker forms of the abc conjecture from the theory of Hölder and logarithmic means

In this post (the content of this post is now cross-posted from Mathematics Stack Exchange see below) we denote the radical of an integer $n>1$ as the product of disctinct primes dividing it $$\...
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119 views

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-...
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4answers
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For $x$ irrational, is $a_{n} =\sum_{k=1}^{n}(-1)^{⌊kx⌋}$ unbounded?

For $x$ irrational, define $a_{n} :=\sum_{k=1}^{n}(-1)^{⌊kx⌋}$. Can you prove that $\left\{a_n\right\}$ is unbounded? I feel that it is not easy to treat every irrational $x$. I have asked in S.E. ...
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80 views

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 ...
11
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1answer
381 views

Lindemann theorem for Artin-Hasse exponential

Though the Lindemann--Weierstrass theorem is not known in the $p$-adic settings, its "Lindemann" part -- the transcendence of $\exp(a)$ for algebraic $a$ with $0<|a|_p<p^{-1/(p-1)}$ -- was shown ...
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1answer
212 views

Distribution of partial quotients of $\sqrt{d}$ with $X < d \leq 2X$

Recall that for a real number $\alpha$, it has a continued fraction expansion usually written as $$\displaystyle \alpha = [a_0; a_1, a_2, \cdots].$$ Moreover, $\alpha$ is rational if and only if its ...
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1answer
220 views

How often a random walk with irrational increments is close to 0?

Let $\omega$ be an irrational number, and $X$ a random variable taking values $1,-1,\omega,-\omega$ each with probability $1/4$. Let then $X_i$ be iid variables with the same law as $X$ and $S_n=\sum_{...
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0answers
161 views

Action on the upper half plane of the double coset of an irrational matrix in $\mathrm{SL}_2(\mathbb{R})$

Consider the action of $\mathrm{SL}_2(\mathbb{R})$ on the upper half plane $\mathbb{H}$ by Möbius transformations. Denote by $\Gamma$ the group $\mathrm{SL}_2(\mathbb{Z})$. It is known that if $z \in ...
2
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0answers
113 views

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 ...
6
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1answer
399 views

Sign in Dirichlet's approximation theorem

Fix $\alpha \in \mathbf{R}$. The classical Dirichlet's approximation theorem states there exist infinitely many rationals $p/q$ such that $$ \left|\alpha-\frac{p}{q}\right|<\frac{1}{q^2}. $$ ...
2
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2answers
216 views

Approximation of a square with an irrational arithmetic progression

Let $\alpha \in \mathbb{R}\setminus \mathbb{Q}$ be irrational. Does the arithmetic progression $(n\alpha )_{n\in\mathbb{N}}$ becomes arbitrarily close to squares? More precisely, what can be said ...
4
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1answer
135 views

On the set of good approximators in the sense of Dirichlet's theorem

This question came up when thinking about an older question that hasn't been answered as of now. Let $\mathbb{N}$ be the set of positive integers. If $\alpha\in\mathbb{R}$, we say $q\in\mathbb{N}$ is ...

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