Questions tagged [diophantine-approximation]

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2 votes
0 answers
112 views

A sequence linked to irrationality

Let $0 < c < 1$ be a real number and $ x \in \mathbb{R}$. We define the sequence $(u_n)_{n \in \mathbb{N}}$ by : $$u_0 = x$$ $$ \mathrm{If}, u_n \le c, \mathrm{then}, u_{n + 1} = u_n + (1 - c) $$...
8 votes
0 answers
121 views

Finding a rational point of large height on an elliptic curve knowing a real approximation

Let $y^2=x(x^2+n)$ be an elliptic curve with $n\in\Bbb Z$ (the same question can of course be asked for a general e.c). I know (e.g. it has rank 1) that there exists a nontrivial rational point $(r,s)$...
4 votes
1 answer
387 views

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{...
3 votes
1 answer
249 views

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}{...
2 votes
0 answers
112 views

The connection of Faltings height and Tate module

Suppose $K$ is a number field, $S$ a finite set of places of $K$, and $A$ is an abelian scheme over $\mathcal O_{K,S}$. I want to ask is there some connections between the Faltings' height $h_F(A)$ ...
7 votes
1 answer
364 views

Bounding the growth of rational bivariate polynomials from below

The following question is an attempt to find a lower bound for the value of a polynomial at integer points. It is something that I originally thought about while trying to understand how it would be ...
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}...
3 votes
0 answers
79 views

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 $T$-...
-5 votes
1 answer
558 views

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 ...
0 votes
1 answer
139 views

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 ...
9 votes
3 answers
3k views

Simultaneous diophantine approximation

Let $r(x)$ be the function $x$ mod $1$, i.e. $x$ minus its floor. Now let $m$ be a given positive integer, and $c$ a vector in $\mathbb{R}^m$ whose components are linearly independent over $\mathbb{Q}...
1 vote
0 answers
105 views

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 ...
0 votes
0 answers
54 views

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{...
1 vote
1 answer
107 views

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 votes
0 answers
131 views

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 ...
2 votes
1 answer
184 views

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 $\...
3 votes
0 answers
100 views

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 ...
10 votes
1 answer
705 views

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 ...
142 votes
7 answers
14k views

Source and context of $\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 dx/(1+x^2)$?

Possibly the most striking proof of Archimedes's inequality $\pi < 22/7$ is an integral formula for the difference: $$ \frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2}, $$ where the ...
5 votes
1 answer
258 views

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 ...
1 vote
0 answers
85 views

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 ...
1 vote
0 answers
284 views

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 $...
5 votes
1 answer
237 views

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 $$\displaystyle |y^2 - x^3 - Ax - B| = O(|x|^{1/2 + \theta}).$$ ...
5 votes
1 answer
717 views

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 $$\...
2 votes
1 answer
182 views

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:...
1 vote
1 answer
228 views

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.
0 votes
0 answers
107 views

$\log$-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+...
6 votes
0 answers
263 views

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 ...
2 votes
0 answers
134 views

"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\...
1 vote
0 answers
237 views

Norm related to diophantine approximation?

I'm trying to read this paper: Bourgain, J.; Jitomirskaya, S., Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential, J. Stat. Phys. 108, No. 5-6, 1203-1218 (2002), ...
0 votes
0 answers
74 views

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: $abc$-...
2 votes
0 answers
86 views

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 $\...
7 votes
2 answers
720 views

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 ...
0 votes
0 answers
108 views

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 ...
3 votes
1 answer
104 views

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 $$\...
2 votes
1 answer
211 views

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 ...
10 votes
1 answer
602 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 ...
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 ...
5 votes
1 answer
192 views

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 \...
24 votes
1 answer
2k views

Irrationality measure of log(2)/log(6)

As part of my Phd thesis on aperiodic Wang tilings, I've discovered I need a bound on the irrationality measure of $\gamma = \log 2/\log 6$. That is, I am looking for an upper bound on the quantity $...
3 votes
1 answer
164 views

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 vote
0 answers
87 views

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 ...
1 vote
0 answers
74 views

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}{...
-2 votes
1 answer
149 views

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&...
8 votes
1 answer
300 views

Are there partially algebraic Hecke characters?

$\newcommand{\C}{\mathbb{C}} \newcommand{\U}{\mathbb{U}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}}$ Let $F$ be a number field. Let $\chi\colon \mathbb{A}_F^\...
2 votes
1 answer
153 views

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 $...
1 vote
0 answers
81 views

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 ...
1 vote
1 answer
152 views

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 || \...
6 votes
1 answer
448 views

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 ...
0 votes
1 answer
87 views

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-\...

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