Questions tagged [diophantine-approximation]
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332
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Exponential of Liouville Numbers
By Mahler classification of Transcendental real numbers (into the sets of $S$-, $T$- and $U$-numbers), we know that
Any Liouville number is a $U$-number.
$\log \alpha$ is either an $S$- or a $T$-...
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0
votes
1
answer
126
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Diophantine equations involving recurrence sequences
I am working on a Diophantine equation by using transcendental and reduction methods given by Baker and Davenport. However, when I read some papers i don't understand the reduction step, for example ...
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1
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Central limit theorem for irrational rotations
Let $\alpha$ be an algebraic integer of modulus 1, and $ R_\alpha z=\alpha z$. Is
$$\lim_{n\to\infty}\frac{\log|\sum_{k=1}^n \Re R_\alpha^k z|}{\log n}=\frac12$$ for all $z\in S^1$?
Birkhoff's ergodic ...
- 2,075
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100
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Diophantine approximation away from $0$
Let $\alpha$ be a real irrational algebraic number. The now-classic Thue-Siegel-Roth theorem asserts that for any $\varepsilon > 0$ there exists a positive number $c = c(\alpha, \varepsilon)$ such ...
- 24.9k
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Diophantine-like approximation of dynamical subsystems
For $\alpha\in [0,1)$ irrational we know that there exists a sequences $\{ q_n \}_{n=1}^\infty\subseteq \mathbb{N}$ and $\{ p_n \}_{n=1}^\infty\subseteq \mathbb{Z}$ such that
$$ \Big\vert \alpha-\frac{...
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1
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1
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97
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Simultaneous rational approximations of multiples of the golden ratio
My question concerns potential simultaneous rational approximations of irrational numbers.
Let $\alpha = \frac{1 + \sqrt{5}}{2}$ be the golden ratio, and $k \in \mathbb{N}$ a positive integer. In what ...
- 2,606
2
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0
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131
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A problem raised by Roth's theorem and the notion of approximation exponent
Roth's theorem states that every algebraic irrational has approximation exponent equal to $2$. It follows from Theorem 1 of https://arxiv.org/abs/math/0406300 that the approximation exponent of an ...
- 4,093
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1
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175
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Extreme case bounds on Diophantine approximation
I am wondering about the possible best case approximation and worst case approximation of irrational numbers. I think the appropriate formulation is whether there are functions $\hat{b}(q)$ and $\...
- 471
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1
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93
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Mixed integer program and continuous Diophantine approximation
Let $n\in\mathbb{N}$ such that $n\geq 2$ and let $0<r<1$ be a real number. We wish to solve the following problem.
$$\min_{(t,(z_j)_{j=2}^n) \in \mathbb{R}\times \mathbb{Z}^{n-1}} t$$
subject to ...
10
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1
answer
692
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Continuous variant of the Chinese remainder theorem
Let $x_1,x_2,\ldots, x_k \in [0,1]$ be irrational numbers.
I'm interested in what, if anything, can be said about the values $\{nx_i \bmod 1: n\in \mathbb{N}\}$.
Specifically, I'm interested if there ...
- 359
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225
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Diophantine equations involving the difference between perfect square and perfect cube
(a) Do there exist infinitely many triples $(x,y,z)$ of integers with $z\neq 0$ such that
$$
z(x^3-y^2) = x+1.
$$
(b) The same question for
$$
z(x^3-y^2) = y+1.
$$
In other words, are there infinitely ...
- 4,813
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0
answers
81
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Motivation for the Weil function for a Cartier divisor
I'm not sure if this is the right place for a question like this.
In Diophantine approximation, on a complex variety $X$ there is a notion of a Weil function for a Cartier divisor $D$ on $X$ which is ...
- 855
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283
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Can we avoid all algebraic numbers?
We say a polynomial $p$ in $n$ variables degree at most four, and coefficients $-1,0,1$ is $n$-plain.
We say $x$ is an $n$-plain algebraic number if there exists an $n$-plain polynomial $p$ such that
$...
- 469
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0
answers
90
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Diophantine approximation with restricted denominators and prescribed irrationality measure
While studying analytic aspects of turbulence of fluids and waves, I came across very interesting questions in Diophantine approximation, a beautiful topic that I am not an expert in.
The question ...
5
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Integral points near elliptic curves
This question is an extension of my earlier question here, answered by Noam Elkies.
Let $A,B \in \mathbb{Z}$. Consider the inequality
$$\displaystyle |y^2 - x^3 - Ax - B| = O(|x|^{1/2 + \theta}).$$
...
- 24.9k
5
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1
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Upper bound for Hall's conjecture on separation of squares and cubes
Hall's (weak) conjecture is the statement that for all $\varepsilon > 0$ there exists a positive number $c(\varepsilon) > 0$ such that for all $x,y \in \mathbb{Z}$ with $y^2 \ne x^3$, that
$$\...
- 24.9k
2
votes
1
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180
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Simultaneously approximating all $x \in [0,1]$ with fractions of bounded denominator
Dirichlet's theorem says that all numbers $x\in [0,1]$ can be approximated by infinitely many fractions $p/q \in \mathbb{Q}$ with error $|x - p/q| \le 1/q^2$.
I am interested in the following question:...
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1
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The liminf of an expression involving an irrational rotation
Let $0 < a < 1$ be an irrational number. Is it true that
$$\liminf_{n \in \mathbb N, n \to \infty} n \{na\} = 0?$$
Note: Here $\{\cdot\}$ denotes the fractional part.
- 3,802
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107
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$\log$-classes of irrationals
Let $\mathbb{N}$ denote the set of non-negative integers. For $A\subseteq \mathbb{N}$ we let the (upper) density of $A$ be defined by $d^+(A) = \lim\sup_{n\to\infty} \frac {|A\cap \{0,\ldots, n\}|}{n+...
- 44.4k
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"Almost rational" irrational
This is a follow-up to an older question.
Let $r\in \mathbb{R}\setminus\mathbb{Q}$, let $\mathbb{N}$ denote the set of non-negative integers and let $\mathbb{N}_+=\mathbb{N}\setminus\{0\}$. For $n\in\...
- 44.4k
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Extreme elliptic curves from good $abc$-triples
It is a well-known fact that the $abc$-conjecture of Masser and Oesterle and Szpiro's conjecture are equivalent. For the convenience of the reader I will write down the statements for both:
$abc$-...
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On fractional parts and Behrend’s construction
Given $\theta \in \Bbb{T}^D := \Bbb{R}^D/\Bbb{Z}^D$, write $f_\theta$ for the homomorphism from $\Bbb{Z}\to \Bbb{T}^D$ induced by $1\mapsto \theta$.
For $x\in \Bbb{T}^D$, let $||x||$ be the smallest $\...
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2
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Well known applications of Roth's theorem
Roth's theorem in Diophantine approximation (1955) is a well known milestone. It has been generalised in the case of number fields for simultaneous approximations considering several places.
It is an ...
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107
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Improving Diophantine approximation by rescaling
Let $\lambda\in(0,1)$ be an irrational number such that its continued fraction expansion is bounded (for example, an irrational quadratic number, whose continued fraction is periodic). It is known ...
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1
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Dirichlet's theorem with an arbitrarily small constant for algebraic numbers of degree $d \geq 3$
Dirichlet's theorem on diophantine approximation asserts that, for every irrational real number $\alpha$, there are infinitely many rational numbers $p/q$ with $\gcd(p,q) = 1, q > 0$ such that
$$\...
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Is there an adelic proof of Gallagher's ergodic theorem?
Gallagher's ergodic theorem in Diophantine approximation states that the approximation of real numbers by rationals obeys a striking 'all or nothing' behaviour.
For the sake of fixing notation, I'll ...
- 1,394
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1
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views
Baker's theorem for integer combinations of logarithms of integers?
Baker's theorem in transcendental number theory states that
$$
\left|\beta_0 + \sum_{i=1}^n \beta_i \log \alpha_i\right| > H^{-C}
$$
where
$\beta_0, \ldots, \beta_n$ are algebraic numbers, not ...
- 856
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votes
1
answer
210
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Measuring how "close" $\alpha\in[0,1]\setminus\mathbb{Q}$ is to being rational
Let $\mathbb{N}_+$ denote the set of positive integers and let $\mathbb{N}_0 = \mathbb{N}_+\cup\{0\}$. Fix $\alpha\in[0,1]\setminus \mathbb{Q}$. For $n\in\mathbb{N}_+$ we let the approximation radius ...
- 44.4k
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Does there exist a sequence $(x,y) \in \mathbb{Z}^2$ such that $|\alpha x - y| \sqrt{x^2 + y^2}$ approaches a given real number?
Let $\alpha > 0$ be a real irrational algebraic number and $c > 0$.
I am interested in the following question.
Does there exist a sequence $(x_i,y_i) \in \mathbb{Z}^2$ such that
$$
\lim_{i \...
- 3,517
3
votes
1
answer
149
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The growth of certain continued fractions
I was recently looking into an old problem of Hardy which studies the distribution of integers of the form $2^a 3^b \leq x$, where $a,b\geq 0$. Letting $N(x)$ denote the number of pairs $(a,b)$ ...
- 1,708
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Almost modularity of Belyi curves and etale fundamental group of non-Belyi curves
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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0
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72
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Liouville numbers with some "special" convergents
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
$$
0<\left|\ell-\frac{p_n}{q_n}\right|<\dfrac{1}{...
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1
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On a criterion for unimodular matrix [closed]
A matrix $$\begin{bmatrix}w &x \\\ y &z\end{bmatrix}\in\mathbb Z^{2\times 2}$$ is unimodular if $$|wz-xy|=1$$ holds.
Is there a parametrization of such matrices with $2wy>(wz+xy)$ and $2xz&...
- 13.5k
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votes
1
answer
151
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Almost Diophantine approximation
We have an algebraic number $a$ and a real number $b$. Can the following inequality have infinitely many solutions for $n \in \mathbb{N}$?
$$ \{an\} \in [b - \frac{1}{2^n}, b + \frac{1}{2^n}] $$
Here $...
- 123
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0
answers
68
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Dyson's lemma implies index is small (in proving Roth's theorem)
I am reading the proof of Roth's theorem in Hindry-Silverman's book. In there they used Roth's lemma. I think it is well known that the step of Roth's lemma could be replaced by Dyson's lemma to show ...
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1
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149
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Distribution of $\alpha n^2/q$ modulo $1$?
Let $0 \neq \alpha \in [0,1]$ and $q$ a positive integer.
Let $||.||$ denote the distance to the closest integer and define
$$
N_i(q) = \sum_{ \substack{ -q/2 \leq n \leq q/2 \\ \frac{i}{q} \leq || \...
- 3,517
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1
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Equidistribution modulo 1
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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Simple estimation of difference of powers of 2 and powers of 3?
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
$$ \left| \frac{\log 2}{\log 3} - \frac{...
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votes
1
answer
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Measuring the quality of real approximation
Let $r\in [0,1]\setminus\mathbb{Q}$, let $\mathbb{N}$ denote the set of non-negative integers and let $\mathbb{N}_+=\mathbb{N}\setminus\{0\}$. For $n\in\mathbb{N}_+$ let $$\alpha_r(n)=\min\{\big|r-\...
- 44.4k
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votes
1
answer
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Markov constant of $\pi$
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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A question on multiplicative diophantine approximation
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 $(\...
- 11
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answers
64
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Stability of successive minima with respect to the metric on the space of lattices
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 ...
- 1,543
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0
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149
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Uniform distribution mod $1$ vs independence of random variables
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 ...
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2
answers
125
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The range of each of successive minima for all unimodular lattices
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 ...
- 1,543
9
votes
1
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544
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How far away can we get by multiple rounding and unit change?
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 ...
- 28.4k
2
votes
1
answer
185
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When does a random trigonometric sum approximate $1$?
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
$$
\frac 1n\...
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votes
1
answer
438
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Direct proof that the set of badly approximable numbers have full Hausdorff dimension without using Schmidt games
A badly approximable number is an $x$ for which there is a positive constant $c$ such that for all rational $p/q$ we have
$$\left|{ x - \frac{p}{q} }\right| > \frac{c}{q^2} \ . $$
The set of badly ...
- 1,543
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0
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379
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Simple Diophantine equation
Are there any solutions in positive integers of
$x^3 + 1 = (x - k) y^3$?
The closest I can get is
$19^3 + 1 = 20 \times 7^3$,
but $20\gt 19$ so it just misses!
For the related
$x^3 - 1 = (x - k) y^3$,...
- 595
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votes
0
answers
248
views
Baby $abc$ conjecture for $n$-th roots
Is there any progress on a “baby $abc$ conjecture” where you restrict attention to rational approximations of $n$-th roots?
Let $r/s$ be a very close approximation to $(t/u)^{1/n}$, so that
$$
|u\cdot ...
- 595
3
votes
1
answer
233
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 ...
- 3,517