Questions tagged [diophantine-approximation]
The diophantine-approximation tag has no usage guidance.
334
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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
$$
|u\cdot ...
3
votes
1
answer
238
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number of integers $n$ with $\|n \alpha \|$ small?
Let $\alpha \in \mathbb{R}$ and $N$ a positive integer. I am interested in the quantity
$$
D(\alpha, N) := \# \{ n \in [1, N]: \| n \alpha \| < 1/N \},
$$
$\| x \|$ denotes the distance to the ...
3
votes
2
answers
116
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The closure of the orbit of an irrational grid contains the fiber
Let $G=\text{SL}(d,\mathbb R)$ and $\Gamma = \text{SL}(d,\mathbb Z)$. The homogeneous space is identified with the space of unimodular lattices, denoted $X_d$. Let $Y_d$ denote the space of unimodular ...
5
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Does the equation $a^b+b^c+c^a=d^e$ have solutions in $\mathbb {N}$
Here $a,b,c,d,e$ are distinct and all greater than $1$.
This question was formerly posted on Math.Stackexchange, precisely here,
but seems to be more general than some other tough number theory ...
3
votes
1
answer
247
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Distance formula for continued fractions
In the book Neverending fractions from Borwein, van der Poorten, Shallit and Zudilin, there is the so called distance formula (Theorem 2.45, p. 43) stated:
$$\alpha_1\alpha_2\cdot...\cdot\alpha_n=\...
5
votes
1
answer
507
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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 ...
2
votes
1
answer
293
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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
$$\|Aq-p\|^m < \frac{1}{T}, \|q\|^...
6
votes
1
answer
264
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A property of rapid sequences of natural numbers
$\newcommand{\IR}{\mathbb R}$
$\newcommand{\IT}{\mathbb T}$
$\newcommand{\w}{\omega}$
$\newcommand{\e}{\varepsilon}$
Taras Banakh and me proceed a long quest answering a question of ougao at ...
-1
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1
answer
141
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A pathological (?) function involving powers
This is inspired by a recent math.SE question.
Given that mathematicians like to come up with theoretical constructs which do not necessarily always have any practical purpose (but sometimes provide ...
1
vote
1
answer
185
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Bounding the fractional parts of the $p^{\text{th}}$ roots of $n,n^2,...,n^{p-1}$
EDIT (August 9, 2021): I would like to ask a more general question. The original question that was fully answered is below the line.
For a positive real number $x$, denote the fractional part $x-[x]$ ...
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324
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Given $2^n - 1 \mid 3^m - 1$, how large must $m$ be compared to $n$?
Let $m,n$ be natural numbers such that $2^n - 1 \mid 3^m - 1$. By results from Bugeaud-Corvaja-Zannier, say Theorem 3 of this paper , we know that for any constant $C > 0$ we must have $m > Cn$ ...
2
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0
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139
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The analogue of Liouville's inequality in several variables
Liouville's Theorem in Diophantine approximation asserts that for each irrational algebraic number $\alpha$ of degree $d$, we have
$$ |q\alpha - p| \gg_\alpha |q|^{1-d}$$
for all integers $p,q \in \...
3
votes
1
answer
211
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Determine if a 2-variable Diophantine equation has a finite or infinite number of solutions
Do there exist an algorithm, which, given a polynomial $P(x,y)$ with integer coefficients, determines whether Diophantine equation $P(x,y)=0$ has finite or infinite number of integer solutions?
Famous ...
3
votes
1
answer
249
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What is the sequence of badly approximable numbers to omit in Hurwitz' second Theorem for Diophantine Approximation to obtain better constants?
The well known result of Hurwitz on Diophantine approximation says that for any irrational $\alpha$ there are infinitely many integer numbers $p$ and $q$ such that
$$
|\alpha -\frac{p}{q}|<\frac{1}{...
1
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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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145
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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 ...
1
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0
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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'...
3
votes
0
answers
122
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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|...
3
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143
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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
$$ A(\alpha,\beta;\varepsilon) = \{ n \in \mathbb{N} \ : \ \Vert \alpha n \Vert ...
1
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0
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73
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On the degree of irrationality of two irrational numbers and their rational (in)dependence
Let $x$ and $y$ be some irrational numbers. If the degree of irrationality of $x$ is the same as that of $y$, is it necessarily the case that $x$ and $y$ are rationally dependent ?
ADDENDUM: What if $...
1
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1
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Bounded, aperiodic irrationals with bounded, aperiodic sum
If $q = [q_0;q_1 \dots]$, say $q_i$ is the $i$-th partial quotient of $q$. My question is the following:
Can one construct an explicit example of irrational $r,s > 0$ such that
$\{ 1,r,s\}$ is $\...
4
votes
1
answer
236
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More about Roth's theorem: bound for the constant and multidimensional case
For a real number $x$, we denote
$$ \|x\|=\inf_{m\in {\Bbb Z}}|x+m|.$$
Problem 1:
Roth's theorem states that given any irrational algebraic number $\alpha$ and for any $\epsilon>0$, there exists a ...
1
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0
answers
139
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On the number of asymptotic solutions of the linear Diophantine equation
Given positive integers $a,b,c$ which are pairwise coprime. Let $f(n)$ denotes the number of the non-negative integer solution $(x,y,z)$ to the equation
$$ax+by+cz=n.$$
we have Prove that there exists ...
2
votes
1
answer
308
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Find better than $ 4^n\prod_{k=1}^{n-1}\cos^2(k)\sim e^{o(n)}$
Let $$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(...
1
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2
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267
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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+...
4
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263
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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}...
3
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0
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99
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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 : {...
1
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0
answers
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 $...
0
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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 ...
2
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0
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145
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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 \...
1
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0
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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 ...
6
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0
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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 $(p,q)$ ...
0
votes
0
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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 ...
8
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1
answer
439
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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 ...
0
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0
answers
46
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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
$(p_n,q_n),n\in \mathbb{N}$ ...
5
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0
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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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1
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355
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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 ...
3
votes
1
answer
262
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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 ...
0
votes
0
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81
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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 ...
1
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2
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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 ...
6
votes
1
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588
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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)=...
2
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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 $$\...
2
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0
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139
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Double Diophantine approximation
Let $0 < \alpha < 1$. For any $n$ there is a closest lower Diophantine approximation $\max p / q \leq \alpha$ with integer $0 \leq p < q \leq n$. It can be found efficiently, e.g., with Stern-...
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votes
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For $x$ irrational, is $a_{n} =\sum_{k=1}^{n}(-1)^{⌊kx⌋}$ unbounded?
For $x$ irrational, define $a_{n} :=\sum_{k=1}^{n}(-1)^{⌊kx⌋}$. Can you prove that $\left\{a_n\right\}$ is unbounded?
I feel that it is not easy to treat every irrational $x$.
I have asked in S.E. ...
2
votes
0
answers
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Bounds on convergence of two orbits in the limit set of a Schottky group
Suppose we have two points in the limit set of a Schottky group, $x,y\in \Lambda(\Gamma)$. Consider the orbits of those points under a primitive subset $\Gamma'$ of $\Gamma,$ that is, one that does ...
13
votes
2
answers
712
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Lindemann theorem for Artin-Hasse exponential
Though the Lindemann--Weierstrass theorem is not known in the $p$-adic settings, its "Lindemann" part -- the transcendence of $\exp(a)$ for algebraic $a$ with $0<|a|_p<p^{-1/(p-1)}$ -- was shown ...
3
votes
1
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267
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Distribution of partial quotients of $\sqrt{d}$ with $X < d \leq 2X$
Recall that for a real number $\alpha$, it has a continued fraction expansion usually written as
$$\displaystyle \alpha = [a_0; a_1, a_2, \cdots].$$
Moreover, $\alpha$ is rational if and only if its ...
8
votes
1
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How often a random walk with irrational increments is close to 0?
Let $\omega$ be an irrational number, and $X$ a random variable taking values $1,-1,\omega,-\omega$ each with probability $1/4$. Let then $X_i$ be iid variables with the same law as $X$ and $S_n=\sum_{...
2
votes
0
answers
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Does $\sum_{i\le k}\mathrm{frac}(n\alpha_i)<1$ hold infinitely often?
For each $t \in \mathbf{R}$, let $\mathrm{frac}(t)$ be its fractional part.
Question. Fix reals $\alpha_1,\ldots,\alpha_k \in (0,1)$ such that $\sum_{i\le k}\alpha_i<1$. Do there exist ...
6
votes
1
answer
433
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Sign in Dirichlet's approximation theorem
Fix $\alpha \in \mathbf{R}$. The classical Dirichlet's approximation theorem states there exist infinitely many rationals $p/q$ such that
$$
\left|\alpha-\frac{p}{q}\right|<\frac{1}{q^2}.
$$
...