Questions tagged [diophantine-approximation]
The diophantine-approximation tag has no usage guidance.
338
questions
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Is it possible to approximate irrational by fractions with denominator and numerator odd? [closed]
Suppose $\alpha$ is a positive irrational, and $\epsilon$ is an arbitrary positive real, are there $m,n$(non-negative integers) such that $$|\alpha-(2m+1)/(2n+1)|<\epsilon/(2n+1)?$$
If they exist, ...
- 191
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0
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194
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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 ...
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185
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Asymptotic formula, polynomial, irrational number and uniformly distribution
Problem 1
Given a irrational number $\alpha$ and two polynomials with positive integer coefficients $P(n),Q(n)$, is it possible to get the asymptotic estimate and reasonable error term for:
$$\...
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-2
votes
1
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Inverse to a Diophantine number [closed]
A number $\alpha$ is said to satisfy the Diophantine condition with
exponent $\beta$ iff for some constant $C>0$ the estimate
$$
\left| \alpha - \frac{p}{q} \right| > \frac{C}{q^{2+\beta}}
$$
...
- 49
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450
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Diophantine approximation of algebraic number
An important theorem in Diophantine approximation is the theorem of Liouville:
Liouville Theorem If x is a algebraic number of degree $n$ over the rational numbers then there exists a constant c(x) ...
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2
votes
1
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443
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Counting algebraic points of bounded height
Let $K$ be a number field and $X\hookrightarrow\mathbb P^n_K$ be a projective variety of degree $\delta$ (with respect to universal bundle) and dimension $d$. We denote the set
$$S(X;D,B)=\{\xi\in X(\...
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4
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Is the series $\sum_n|\sin n|^n/n$ convergent?
Problem. Is the series $$\sum_{n=1}^\infty\frac{|\sin(n)|^n}n$$convergent?
(The problem was posed on 22.06.2017 by Ph D students of H.Steinhaus Center of Wroclaw Polytechnica. The promised prize for ...
- 4,435
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0
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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 ...
- 2,199
9
votes
1
answer
260
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Cover of the positive real numbers by intervals
For which real numbers $x$ and $y$ does the following hold?:
$$
\bigcup_{\frac{a}{b} \in \mathbb{Q}^+}
\left[\frac{a}{b},\frac{a}{b}+\frac{1}{a^x b^y}\right]
\ = \ \mathbb{R}^+
$$
- 19.5k
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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 $$v_1=\begin{bmatrix}a_1&b_1\end{bmatrix}\in\Bbb Z_{>0}^2$$
...
- 13.7k
2
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1
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97
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Limits of a quasiperiodic function with two pseudoperiods
Let $\beta$ be a real number such that $\beta^2\notin\mathbb{Q}$. For any smooth function $f$ on $\mathbb{R}$ that decreases sufficiently at infinity, for example a Gaussian function, let us define
$$
...
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10
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226
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Distribution of good diophantine approximations
Let $\langle x \rangle: \mathbb{R} \to (-1/2,1/2]$ be the periodic function with period $1$ which is $x$ for $x \in (-1/2,1/2]$. Is there some function $D(a,b)$ of real numbers $a<b$ such that, for ...
- 151k
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How large are good approximations to irrational numbers?
It is well known that for almost every $c \in \mathbb{R} / \mathbb{Z}$ there exists $(q_n)_{n \geq 1}$ and $(a_n)_{n \geq 1}$ such that $$|c - a_n / q_n| \leq 1/ q_n^2,$$ where $q_n < q_{n+1} \leq ...
- 2,283
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446
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Sequential addition of points on a circle, optimizing asymptotic packing radius
Suppose I have to put $N$ points $x_1, x_2, \ldots, x_N$ on the circle $S^1$ of length 1 so as to achieve the largest minimum separation (packing radius). The optimal solution is the equally spaced ...
- 5,926
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461
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A naive diophantine approximation question
Let $\alpha$ be a positive real number (bigger than one, and irrational if it matters) (the one I am secretly thinking of is $\varphi,$ the golden ratio). I want to know, given an $\epsilon>0,$ ...
- 95.6k
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Growth of a linear recurrent sequence
Consider the sequence defined by $a_0=a_1=1$ and $a_n=2a_{n-1}-3a_{n-2}$ for $n\geq 2$. This is the sequence https://oeis.org/A087455.
I would like to prove that $|a_n|>100$ when $n>10$. How ...
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answers
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 ...
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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 ...
- 22.6k
2
votes
1
answer
225
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Bounds on near-zero integer linear combinations of numbers linearly independent over $\mathbb{Q}$
Let $\alpha_1, \alpha_2, \dots$ be an infinite sequence of real numbers such that any finite subset is linearly independent over $\mathbb{Q}$. Let $f(N)$ be the number of tuples $(m_1, \dots, m_N)$ ...
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Simultaneous Diophantine approximation of $\sqrt{2}$ and $\sqrt{2\pm \sqrt{3}}$
By using the LLL algorithm, I tried to find the best simultaneous Diophantine approximation of the three numbers $\sqrt{2} $ and $ \sqrt{2 \pm \sqrt{3}} $. I was expecting that to get a precision of $\...
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Simultaneous Diophantine approximation in the non-generic case
Suppose we have $n$ irrational numbers $\{ x_1, x_2, \ldots, x_n \}$. For a generic set of such numbers, we have the well-known theorem that there exist infinitely many integers $q$ such that
$$ \...
- 265
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334
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On discrepancy of integer sequences related to Erdos-Turan-Koksma
Assume we have a sequence $a_1,\dots,a_k\in\Bbb N$ with each $a_i\approx n$ where $n$ is some integer.
Suppose there exists an $\eta\in(0,1)$ such that for every $d_1,\dots,d_k\in\Bbb Z$ with $$...
user94040
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2
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508
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Beyond Dirichlet's approximation theorem
I haven't taken any number theory courses but out of curiosity I learned about Dirichlet's approximation theorem. Afterwards, it occurred to me to define the following function $f$ using 'optimal' ...
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votes
1
answer
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Diophantine equation for 2016: triangular $|{\rm GL}_2({\bf F}_q)|$
For a prime power $q$ the group ${\rm GL}_2({\bf F}_q)$ has
$(q^2-1)(q^2-q)$ elements. This happens to be a triangular number for
$q=2$ (being $6 = 1+2+3$), and $-$ more notably, especially this year ...
- 77.5k
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809
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Are there any solutions to the diophantine equation $x^n-2y^n=1$ with $x>1$ and $n>2$?
This problem arose when considering storage of cannonballs in n-dimensional pirate ships, as explained in this MSE post. This MO question can also be reduced to the $n=3$ case. If $x,y$ is a solution ...
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Proving that $(\omega(c_1 x_1^n+\cdots+c_kx_k^n))_{n\ge 1}$ is bounded only if $|x_1|=\cdots=|x_k|$ by the Subspace Theorem
Let $c_1, \ldots, c_k \in \mathbf N^+$ and $x_1,\ldots,x_k \in \mathbf Z \setminus \{0\}$. It is possible to prove by elementary means that $(\omega(c_1 x_1^n+\cdots+c_kx_k^n))_{n\ge 1}$ is a bounded ...
- 9,636
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154
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Square summable sequences associated to Pisot numbers
Given a real number $x$, let $\Vert x\Vert=[x]-x$, where $[x]$ is the nearest integer to $x$.
Suppose $\lambda>1$ is a Pisot number. Let $f(x)=x^k+a_{k-1}x^{k-1}+\ldots+a_0$ be the irreducible ...
- 3,204
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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 $...
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111
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asymptotic behavior of $N_{\mathbb{Q(\alpha)}/\mathbb{Q}} (1-\alpha^n)$
We have an algebraic complex number $\alpha$ such that $|\alpha|=1$ and it is not a root of unity, We also know that $\alpha \in \mathcal{O}_{\mathbb{Q}(\alpha)}$ (ring of integers). I need to study ...
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Vojta's conjecture on the bounded degree algebraic points over projective line?
I want to know the accurate version of the Vojta's conjecture on the bounded degree algebraic points over projective line, which is known as an extension of the Roth's theorem for bounded degree ...
- 81
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0
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147
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Metric extensions of Littlewood's conjecture
Littlewood's conjecture on simultaneous rational approximation to a pair of real numbers,
$$
\liminf_{n \in \mathbb{N}} \, n \cdot \mathrm{dist}(n\alpha,\mathbb{Z}) \cdot \mathrm{dist}(n\beta, \mathbb{...
- 13.7k
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Irrationality measure of the number is itself
Does there exist real number $\theta\in \mathbb{R}$\ $\mathbb{Q}$ such that Irrationality Measure of $\theta$ is itself?
$$\forall \epsilon >0, \exists C>0, \forall(p,q)\in \mathbb{Z^2},\bigg|\...
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2
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925
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Estimate number of solutions in the Roth's theorem
There is a fundamental theorem in Diophantine approximation :
For all algebraic irrational $\alpha$
$$\displaystyle \left \lvert \alpha - \frac{p}{q} \right \rvert < \frac{1}{q^{2 + \...
- 493
2
votes
0
answers
106
views
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
$$
f({\bf a})=\sum_{i=1}^r a_i x_i,\quad {\bf a}\in \{...
- 2,085
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Analog of Baker's theorem on linear combination of $\log a \log b$
Baker's theorem basically says that, given algebraic numbers $a_1,\ldots,a_n$ and $m_1,\ldots,m_n$, if there is no good reason for a linear combination
$$\sum m_i\log a_i$$
to cancel, then it is ...
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Efficient Dirichlet approximation (continued fractions?) over a number field
Is there an efficient algorithm for Dirichlet approximation for a given (high-degree) number field and its ring of integers, perhaps analogous to the Euclidean/continued fractions algorithm for the ...
- 186
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0
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561
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Optimal Diophantine approximation of $\pi$
If the 'optimal' Diophantine approximation of $\pi$ is given by the maximum value of $M=-\log_q(\min_{\forall p \in \mathbb{N}} |\frac{p}{q}-\pi|)$ for $q \geq 2$, what is this value?
- 3,827
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Criterion for irrational numbers of constant type 2
From Kuiper's and Niederreiter's book Uniform distribution of sequences, Ch.2, § 3, I learn that an irrational number $\alpha\in \mathbf{R}\smallsetminus \mathbf{Q}$ is of constant type $\eta$ if ...
- 5,609
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Solving polynomial equations modulo $1$
Let $P\in \mathbb{R}\lbrack x\rbrack$ be given. (In practice, the coefficients could be given as, say, decimals to sufficient precision.) Let $M\geq 1$, and let $I$ be an interval in $\mathbb{R}/\...
- 19.4k
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1
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461
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On property of monic polynomial with integer coefficients
For a monic polynomial with integer coefficients $f$ where $\partial f = 2$, we have
$$
\textrm{inf}(f(x)) > 0 \implies
\textrm{inf}(f(x)) \geq \frac{3}{4} .
$$
Could we generalize this (for ...
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On the construction of a certain sequence of integers
Let $\alpha \in \mathbb{R}$ be a fixed (positive) number. For each $k \in \mathbb{N}$ we choose $\varepsilon_k >0$ with the property that $\lim_k \varepsilon_k =0$.
If $\alpha \in \mathbb{R} \...
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On a sequence of integers
I recall a well-known theorem due to Minkowski.
Theorem. If $\theta$ is irrational and $\alpha$ is not of the form $\alpha = m\theta+n$ for some integers $m$ and $n$, then there are infinitely many ...
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1
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On cluster points of a particular sequence
This is the sequel of a previous question.
Let us consider the sequence
$$
\xi_n = 2n \{n\xi\}-n,
$$
where $\xi>0$ is a given real irrational number and $\{\cdot\}$ is the fractional part.
Do ...
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votes
2
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Precise asymptotic of diophantine approximation
I know that every irrational number $\xi$ can be approximated by rational numbers in such a way that
$$
\left| \xi - \frac{p}{q} \right| \leq \frac{c}{q^2}
$$
for infinitely many choices of $p$ and $q$...
- 459
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votes
2
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340
views
On the density of the sequence $\{n \{n \xi \} \}_n$
I have a question that I can't manage to answer myself. It comes from some work in PDE theory, but it is related to analytic number theory.
Let us say that we have an irrational number $\xi$. The ...
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views
Can two rational rotations $F_2 = \langle A, B \rangle \to SO(3)$ efficiently approximate the $3 \times 3$ identity matrix?
Let $A,B$ be two rational rotations:
$$ A = \left[\begin{array}{rcc} \frac{3}{5} & \frac{4}{5} & 0 \\
-\frac{4}{5} & \frac{3}{5} & 0 \\
0 & 0 & 1 \end{array}\right]
\quad\...
- 22.6k
4
votes
2
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A particular Diophantine approximation of $\pi/2$
I have asked this question in math.stackexchange without any answer, so I have decided to post it here too.
Recently I was playing around with the sequence $$\frac{1}{n\sin(n)},\ n\in\mathbb{N}.$$
...
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2
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1k
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Approximating any integer by multiples of 2 and 3
Given any integer $n$ sufficiently large, I want to prove (or disprove) that there exists another integer $m\ge n$ with the form $m=2^a3^b$ ($a,b$ are no negative integers) such that $m-n=o(n)$, i.e., ...
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0
answers
305
views
Diophantine approximation in the Julia set
Let $f : \mathbb{CP}^1 \to \mathbb{CP}^1$ be a rational map of degree $q > 1$; or just a quadratic binomial $z^2 + c$, if one prefers. The Julia set $J_f$ is the closure of the repelling periodic ...
- 13.7k
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votes
3
answers
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Does $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converge to zero when $\alpha$ is irrational?
I came across a problem concerning about the convergence of products. I wonder if the complex series $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converges to zero when $\alpha$ is irrational. Of course, ...
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