Questions tagged [diophantine-approximation]

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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 ...
Joe Shipman's user avatar
3 votes
1 answer
238 views

number of integers $n$ with $\|n \alpha \|$ small?

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 ...
Johnny T.'s user avatar
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3 votes
2 answers
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The closure of the orbit of an irrational grid contains the fiber

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 ...
No One's user avatar
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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 ...
Derek Luna's user avatar
3 votes
1 answer
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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=\...
Marcus's user avatar
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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 ...
BPP's user avatar
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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\|^...
No One's user avatar
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6 votes
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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 ...
Alex Ravsky's user avatar
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-1 votes
1 answer
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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 ...
Wolfgang's user avatar
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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 $x-[x]$ ...
Jens Reinhold's user avatar
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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 $m > Cn$ ...
user47437's user avatar
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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 \...
Jakub Konieczny's user avatar
3 votes
1 answer
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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 ...
Bogdan's user avatar
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1 answer
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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}{...
Thomas Lachmann's user avatar
1 vote
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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 ...
roydiptajit's user avatar
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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 ...
user avatar
1 vote
0 answers
199 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'...
Adam's user avatar
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0 answers
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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|...
Jakub Konieczny's user avatar
3 votes
0 answers
143 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 ...
Jakub Konieczny's user avatar
1 vote
0 answers
73 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 $...
PRIMES is in P.'s user avatar
1 vote
1 answer
152 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 $\...
Descartes Before the Horse's user avatar
4 votes
1 answer
236 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 ...
Isomorphism's user avatar
1 vote
0 answers
139 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 ...
math110's user avatar
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2 votes
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308 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(...
Pascal's user avatar
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1 vote
2 answers
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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+...
admissiblecycle's user avatar
4 votes
1 answer
263 views

The Hausdorff codimension of singular matrices vs. the Hausdorff codimension of points with divergent trajectories

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}...
No One's user avatar
  • 1,545
3 votes
0 answers
99 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 : {...
coudy's user avatar
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1 vote
0 answers
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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 $...
Randall Fairman's user avatar
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0 answers
69 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 ...
katago's user avatar
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2 votes
0 answers
145 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 \...
katago's user avatar
  • 543
1 vote
0 answers
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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 ...
Stanley Yao Xiao's user avatar
6 votes
0 answers
93 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 $(p,q)$ ...
kaleidoscop's user avatar
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0 votes
0 answers
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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 ...
No One's user avatar
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8 votes
1 answer
439 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 ...
joaopa's user avatar
  • 3,655
0 votes
0 answers
46 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 $(p_n,q_n),n\in \mathbb{N}$ ...
kaleidoscop's user avatar
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5 votes
0 answers
117 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 ...
Jan-Willem van Ittersum's user avatar
3 votes
1 answer
355 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 ...
VS.'s user avatar
  • 1,816
3 votes
1 answer
262 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 ...
kaleidoscop's user avatar
  • 1,268
0 votes
0 answers
81 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 ...
VS.'s user avatar
  • 1,816
1 vote
2 answers
301 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 ...
VoB's user avatar
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6 votes
1 answer
588 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)=...
VS.'s user avatar
  • 1,816
2 votes
1 answer
395 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 $$\...
user142929's user avatar
2 votes
0 answers
139 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-...
Mikhail Tikhomirov's user avatar
26 votes
4 answers
2k views

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. ...
Chennes's user avatar
  • 385
2 votes
0 answers
88 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 ...
user6419's user avatar
  • 431
13 votes
2 answers
712 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 ...
Wadim Zudilin's user avatar
3 votes
1 answer
267 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 ...
Stanley Yao Xiao's user avatar
8 votes
1 answer
260 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_{...
kaleidoscop's user avatar
  • 1,268
2 votes
0 answers
125 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 ...
Paolo Leonetti's user avatar
6 votes
1 answer
433 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 $$ \left|\alpha-\frac{p}{q}\right|<\frac{1}{q^2}. $$ ...
Paolo Leonetti's user avatar

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