# Questions tagged [diophantine-approximation]

The diophantine-approximation tag has no usage guidance.

342
questions

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### Diophantine approximation for polynomials in a single variable

Let $a_0, a_1, \cdots, a_n$ be non-zero algebraic numbers. Consider the polynomial
$$\displaystyle f(x) = a_n x^n + \cdots + a_1 x + a_0.$$
For a real number $\alpha$, put $\lVert \alpha \rVert$ for ...

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1
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### How to show the geodesic orbit of a badly approximable number are/are not homogeneously equidistributed on its orbit closure?

Let $x\in \mathbb R$ be a badly approximable number. By Dani's correspondence, $g_t u_x \mathbb Z^2$ is bounded away from the cusp in $X=\text{SL}(2,\mathbb R)/\text{SL}(2,\mathbb Z)$, identified with ...

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### Enquiry on irrationality measures

Suppose $a,b$ are positive irrational numbers and let $a_0, b_0$ be their corresponding irrationality measures. Define $c=a/b$ and $d=a-b$. Are the respective irrationality measures of $c$ and $d$ ...

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### Exact approximation in $p$ adic

Given a non increasing function $\psi$ the $\psi$ approximable points in $\mathbb{R}^n$ is defined as
$W(\psi)=\{x\in\mathbb{R}^n:|qx-p|<\psi(q)\}$ for infinitely many $(q,p)\in \mathbb{Z}^m\times\...

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### Characteristics of numbers that satisfy inequality related to diophantine approximation with perfect squares

This question has come up in my algorithms and physics research. I apologize if this is very basic, but I am new to number theory and it seems this is a number-theoretic question. What can we say ...

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158
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### Cardinality of the set $\#\{ 1 \leq n \leq N: \| \alpha n^2/N \| < 1/N \}$

Let $\alpha \in I$ where $I$ is some closed interval that does not contain $0$.
I am interested in upper bound for
$$
M(\alpha) = \#\{ 1 \leq n \leq N: \| \alpha n^2/N \| < 1/N \}
$$
where $N$ ...

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637
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### Consecutive integers of the form $2^a 3^b 5^c$

Let $\mathcal{N}$ denote the set of all products of (powers of) $2,3$ and $5$:
$$ \mathcal{N} = \{ 2^a 3^b 5^c \ : \ a,b,c \geq 0 \} \subset \mathbb{N}.$$
We use the elements of $\mathcal{N}$ to ...

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59
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### Aligning frequencies

Let $\omega_1, \omega_2, \dots, \omega_n$ be frequencies between $1$ and $\log n$. I would like to find an upper bound for a point $t$ that align these frequencies up to a small error $\delta$, that ...

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### Root separation for polynomials of bounded height

Consider integer polynomials $p$ of degree $\leq d$ and height $\leq H$, irreducible over $\mathbb{Q}$. The separation $\text{sep}(p)$ of $p$ is defined as the minimum absolute difference between any ...

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### 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 :
$$u_0 = x$$
$$ \mathrm{If}, u_n \le c, \mathrm{then}, u_{n + 1} = u_n + (1 - c) $$...

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172
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### 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 $(r,s)$...

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### 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 $h_F(A)$ ...

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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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### 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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578
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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 ...

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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 ...

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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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118
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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 ...

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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 ...

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votes

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198
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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 $\...

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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 ...

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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 ...

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0
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89
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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 ...

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285
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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
$...

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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 ...

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247
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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
$$\displaystyle |y^2 - x^3 - Ax - B| = O(|x|^{1/2 + \theta}).$$
...

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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
$$\...

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1
answer

185
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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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229
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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.

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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+...

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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\...

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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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### 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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### 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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### 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 ...

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### 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 ...

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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 ...

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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 \...

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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 $(a,b)$ ...

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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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### 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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### 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&...

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1
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158
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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 $...

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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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173
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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 || \...

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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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1
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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-\...

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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 ...