Questions tagged [diophantine-approximation]
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383 questions
Score of 7
2 answers
737 views
"Maximally irrational" members of $\mathbb{R}\setminus\mathbb{Q}$
Motivation. Irrationals such that the denominators of the rational approximation sequence grows quickly can be seen to be "easily approximable" by rationals. Conversely, if the denominator ...
- reputation score 58131
Score of 9
2 answers
963 views
Two irrationals with the same approximation sequence
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 $n\in\mathbb{N}_+$ let $$\alpha_r(n)=\min\left\{...
- reputation score 58131
Score of 0
0 answers
75 views
Self-intersections of a curve built from Birkhoff sums of {nα} − ½ over an irrational rotation
my name is nevin.
For irrational α and a constant c > 0, define θ_k = c·Σⱼ₌₁ᵏ ({jα} − ½), and the planar walk γ(N) = Σₖ₌₁ᴺ (cos θ_k, sin θ_k).
Numerically, the number of self-intersections of γ up ...
- reputation score 19
Score of 3
1 answer
288 views
Measuring the approximability of irrationals by fractions [closed]
$\newcommand\irr{\text{irr}}$For any $r\in \mathbb{R} \setminus \mathbb{Q}$ and $n\in\mathbb{N}_+:=\mathbb{N}\setminus\{0\}$, let $q_n(r) = \min\{|r - \frac{a}{n}|: a\in\mathbb{Z}\}$. Note that $$0 &...
- reputation score 58131
Score of 3
2 answers
334 views
Density of good approximations of irrational torus rotations
Given an irrational rotation $\omega \in \mathbb T^d$, we may define the set
$$S_\gamma = \{m \in \mathbb Z^d: \|\langle \omega,m\rangle\|_{\mathbb T} < |m|^{-\gamma}\},$$
and the densities
$$A_\...
- reputation score 83
Score of 3
0 answers
165 views
lower bound for $\| x^{1/2}\|$? --reference request
Let $\| z\|$ denote the distance from $z$ to the closest integer.
I am looking for a reference for a lower bound for
$$
\min_{ B < x < 2B, x \neq \square} \| x^{1/2}\|
$$
for $B \gg 1$. I ...
- reputation score 3949
Score of 2
0 answers
143 views
Is $D=2/7$ an exceptional element of $S_1(4)\cap(1/4,1/2]$? (Jain–Kravitz relative Lonely Runner spectra)
In Jain–Kravitz, Relative Lonely Runner spectra, for an integer velocity vector
$v=(v_1,\dots,v_n)$ the maximal-loneliness deficit is
$$D(v)=\tfrac12-\max_{t\in\mathbb R}\ \min_{i:\,v_i\neq0}\ \|v_i t\...
- reputation score 106
Score of 0
0 answers
101 views
Diophantine exponents for interval exchanges and lamination languages
$\DeclareMathOperator\ice{ice}\DeclareMathOperator\sice{sice}\DeclareMathOperator\ind{ind}\DeclareMathOperator\Dio{Dio}\DeclareMathOperator\len{len}$
Context
Let $(X, \sigma)$ be a minimal subshift ...
- reputation score 136
Score of 3
1 answer
279 views
Maximizing the distance from a point to a line with slope $k$ on a 2-torus $\mathbb{T}^2$
Let $\mathbb{T}^2 = [0,1) \times [0,1)$ be the standard 2-dimensional flat torus. Fix a point $a \in \mathbb{T}^2$. Consider a line passing through the origin with slope $k$. Let $D(k, a)$ denote the ...
- reputation score 1179
Score of 22
2 answers
1456 views
What is a number whose Lagrange number is Freiman's constant?
For any irrational $x \in \mathbb{R}$ we define its Lagrange number to be the supremum of real numbers $c$ such that
$$ \displaystyle{ \left| \frac{p}{q} - x \right| < \frac{1}{cq^2} }$$
has ...
- reputation score 25467
Score of 14
1 answer
555 views
What is the liminf of $\|d\sqrt{2}, d\sqrt{3}, d\sqrt{6}\|_2 \cdot d^{1/3}$?
What is the liminf of $\|d\sqrt{2}, d\sqrt{3}, d\sqrt{6}\|_2 \cdot d^{1/3}$?
Background. For a positive integer $d$, define the scaled approximation error
$$f(d) = \sqrt{\|d\sqrt{2}\|^2 + \|d\sqrt{3}\|...
- reputation score 271
Score of 11
0 answers
565 views
Do there exist sequences with $\liminf x_{n+1}/x_{n}=3$ for which no $\alpha, \varepsilon$ keep $\{\alpha x_n\}$ in $[\varepsilon,1/2]$?
Background
Let $(x_n)_{n=1}^{\infty}$ be a sequence of positive integers and let $$\liminf_{n \to \infty}\frac{x_{n+1}}{x_n}=\lambda.$$ Consider the following property: $$(\star)\qquad \exists\,\alpha\...
- reputation score 191
Score of 0
3 answers
540 views
On pure rational numbers [duplicate]
In the literature is there a terminology for the following property of rational numbers;
A rational number in $\mathbb{Q}$ whose reduced form is in the form $\frac{p}{q}$ for two primes $p,q$.
By ...
- reputation score 642
Score of 1
0 answers
121 views
Rational solutions to some inequalities
While working on a problem in Diophantine approximation, I encountered the following seemingly ``simple'' exercise. Despite trying for many days, I have made no progress. I would greatly appreciate ...
- reputation score 59
Score of 10
1 answer
975 views
Status of Mills' constant
There is a rather confusing state of affairs at Wikipedia concerning Mills' constant. The article on formula for primes mentions that It is not known whether it is irrational, but the article on Mills'...
- reputation score 2417
