Questions tagged [diophantine-approximation]

For questions about approximating real numbers by rational numbers.

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Let $x_0 = 3;\ x_{n+1}=3x_n\ $ if $\ \frac{x_n}{2}<1;\ x_{n+1}=\frac{x_n}{2}\ $ if $\ \frac{x_n}{2}>1.\ $ Is $\ \liminf_{n\to\infty} x_n=1?$

This is a natural follow-up question of this previous question of mine. Let $x_0 = 3.$ Let $\ x_{n+1} = 3x_n\ $ if $\ \frac{x_n}{2}<1;\quad x_{n+1} = \frac{x_n}{2}\ $ if $\ \frac{x_n}{2}>1.\quad ...
Adam Rubinson's user avatar
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Philosophy of applying Faltings' product theorem

Faltings' product theorem says that on $\mathbb{P}=\mathbb{P}^{n_1} \times \cdots \times \mathbb{P}^{n_m}$ over $k$ of characteristic $0$. For $\epsilon>0$, $d_1> \cdots >d_m$ decrease ...
finiteness's user avatar
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Show that $| \sum_{0 < \gamma \le T} x^{\rho} | \gg \sum_{0 < \gamma \le T} x^{\beta}$.

In Gonek's paper it is claimed without proof the following: Let $\rho = \beta + i \gamma $ be zeros of Riemann zeta function and $\beta_T = \max_{0 < \gamma \le T} \beta $. Then we deduce from ...
Ali's user avatar
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Approximating $\sqrt2$ by rationals using the Pigeonhole principle

Problem: For any positive integer $m$, I'd like to show that there exist integers $a,b$ satisfying $|a|\leq m$, $|b|\leq m$ and $0< a+b\sqrt{2}\leq \frac{1+\sqrt{2}}{m+2}\,$. Solution attempt: ...
Arthr's user avatar
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4 votes
3 answers
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If $x,y\in\mathbb{N},\varepsilon>0$ then are there infinitely many positive integer pairs $(n,m)$ s.t. $\vert\frac{x^n}{y^m}- 1\vert < \varepsilon?$

Proposition: If $x,y\in\mathbb{N}_{\geq2}$ then for any $\varepsilon>0,$ there are infinitely many pairs of positive integers $(n,m)$ such that $$\frac{\left\lvert y^m-x^n \right\rvert}{y^m} < \...
Adam Rubinson's user avatar
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Examples of specific points $g\in G:=\text{SL}(2,\mathbb R)$ such that the curve $u_tg \Gamma, \Gamma:=\text{SL}(2,\mathbb Z)$ is dense in $G/\Gamma$?

Let $G:=\text{SL}(2,\mathbb R)$, $\Gamma:=\text{SL}(2,\mathbb Z)$ and $u_t=\begin{bmatrix} 1 & t \\ 0 & 1 \end{bmatrix}, t\ge 0$ Although this paragraph is not really needed to answer this ...
taylor's user avatar
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Tighten Corollary of Dirichlet's Simultaneous Approximation Bound (d = 3)

For ease of notation, below when I write $|x| \mod N$, I am referring to the absolute value of the integer with smallest absolute value in the same residue class as $x$. For example $|6| \mod 7$ is $1$...
mathmasterzach's user avatar
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For each $n\in\mathbb{N},$ let $x_n:=\min_{1\leq k < n}\lvert\sin n-\sin k\rvert.\ $ Does $\sum_{n=1}^{\infty} x_n $ converge?

For each $n\in\mathbb{N},$ let $x_n:= \displaystyle\min_{1\leq k < n} \lvert\sin n - \sin k\rvert.\ $ Does $\displaystyle\sum_{n=1}^{\infty} x_n $ converge? Consider instead, $a_1 = 0,\ a_2=1, ...
Adam Rubinson's user avatar
4 votes
1 answer
209 views

The house of an algebraic number

Let $\alpha$ be a non-zero algebraic number of degree $d$. Denote ${\rm den}(\alpha)$ the smallest positive integer $m$ such that $m\alpha$ is an algebraic integer, and ${\rm House}(\alpha)$ the ...
Mystery girl's user avatar
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Dirichlet’s approximation Theorem (Simultaneous version): In case Q is not interger

I’m reading “Diophantine approximation” by W.M.Schmidt. At the Chapter 2, Theorem 1E which is Dirichlet’s approximation Theorem of simultaneous version, He proved the theorem using pigeonhole ...
jihyuk seo's user avatar
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60 views

Finite set all bigger than 1, set of products are dense in some sense

Is there a finite set $S$ of real numbers bigger than $1$, and a positive real number $\alpha$ such that if $P$ = set of finite products, (allow duplication) of elements in $S$ $Q$ = set of ...
imida k's user avatar
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Accumulation point in a topological group of orthogonal matrices over R

I would like to ask your opinion on the point that looks simple. Consider the group of orthogonal matrices of order n over the field R of reals, equipped with the topology induced by the Euclidean ...
Nostromo's user avatar
3 votes
2 answers
185 views

Find all rational numbers $\frac{p}{q}$ such that $|\frac{p}{q}− \frac{17}{12}|< \frac{1}{q^2}$

Find all rational numbers $\frac{p}{q}$ such that $|\frac{p}{q}− \frac{17}{12}|< \frac{1}{q^2}$ Attempt $-\frac{1}{q^2} < \frac{p}{q} - \frac{17}{12} < \frac{1}{q^2}$ $-\frac{1}{q^2} + \frac{...
Mzq's user avatar
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Is there a notion of best $\mathbb{K}$-approximations where $\mathbb{K}$ is an algebraic number field?

A known approximation of $\pi$ is $\sqrt{2}+\sqrt{3}=\color{green}{3.14}\color{red}{6264...}$ But recently I came across a refinement of this approximation $$\pi\approx\sqrt{2}+\sqrt{3}+\frac{\sqrt{2}-...
K. Makabre's user avatar
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Conjecture similar to Dirichlet's approximation theorem but with prime numbers

Dirichlet's approximation theorem states that for any real numbers $\alpha,N$ with $1\leq N,$ there exist integers $p$ and $q$ such that $1\leq q\leq N$ and $\lvert q\alpha-p \rvert \leq \frac{1}{\...
Adam Rubinson's user avatar
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Extreme case bounds on Diophantine approximation

I am interested whether there are functions $\hat{b}(q)$ and $\check{b}(q)$ such that for any $\alpha\in [0,1)\setminus \mathbb{Q}$ we have for only finitely many $q's$ $$ \check{b}(q) \leq \Big\vert \...
Keen-ameteur's user avatar
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7 votes
1 answer
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$\alpha>0;\ a(n+1)\ $ is the least integer $>a(n)$ such that $\sum_{i=1}^{n+1} a(i) < \alpha.$

Let $\alpha>0$ be any positive real number and let $k\ $ be any positive integer satisfying $k > \frac{1}{\alpha}\ $ Consider the integer sequence define by: $a(1)=k;\ $ and for $n\geq 1,\ a(n+1)...
Adam Rubinson's user avatar
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Proof of the convergence part of Jarník's Theorem

Let $\psi: \mathbb{N} \to \mathbb{R}^+$ be some function and denote by $W(\psi)$ the set of numbers $x \in [0,1]$ for which $$\left\lvert x - \frac{p}{q} \right\rvert < \frac{\psi(q)}{q} \quad \...
EndothermicIntegral's user avatar
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1 answer
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Badly Approximable Numbers: Lower Bound for Sum of Reciprocals of Fractional Parts

I am having a bit of trouble proving the following fact that I have seen in a couple of references on Diophantine approximation. Let $||x|| = \min\{|x - p| ~|~ p \in \mathbb{Z}\}$. Recall that a ...
Baldassare Romani's user avatar
1 vote
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Understanding the Proof of Chapter 2, Theorem 8 of Lang's Book on Diophantine Approximations

I am having a bit of trouble understanding the inductive summation step in the proof of Theorem 8 of Chapter 2 in Lang's Introduction to Diophantine Approximations (p. 31). In particular, Lang shows ...
Baldassare Romani's user avatar
7 votes
1 answer
228 views

For $ \{n\pmod{2 \pi}: n \in \{\ 0, 1, 2 ... 2^x\ \} \} $, is it possible to put a bound in terms of $x$ on the smallest difference between elements?

Is it possible to put some sort of bounds on the smallest difference between elements for a limited range of integers $\mod 2\pi$? $$ A =\bigl\{n\pmod{2\pi} : n \in \{\ 0, 1, 2, ... 2^x\ \} \bigl\} $$ ...
EHS's user avatar
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Structure of simultaneous diophantine approximation

Consider an $\alpha \in \mathbb{R} \setminus \mathbb{Q}$. When doing diophantine approximation in one dimension, a result of Lagrange gives the structure of the $p, q \in \mathbb{Z}$ that verify $$ | ...
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Different definitions of "near integral points"

I am reading Javanpeykar's notes on Lang-Vojta's conjecture (https://arxiv.org/pdf/2002.11981.pdf), where I find a definition of "near integral points": $\textbf{Def 1:}$ Let $X \to S$ be a ...
finiteness's user avatar
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The First Main theorem of Nevanlinna theory

I'm currently reading Min Ru's book Nevanlinna theory and its relation to Diophantine approximation and there's a bit that I'm confused. In Theorem A2.3.1, also known as the First Main theorem of ...
oleout's user avatar
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2 votes
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Finding a close integer point to a line with irrational slopes

Let $\alpha$, $\beta$ be two irrational numbers. Is there a good way to find some integer $n,m$ that $|n\alpha-\beta-m|$ is sufficiently small? For example, if $\beta=0$, we know that there exists $n,...
newbie's user avatar
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10 votes
2 answers
199 views

Can we always find arbitrarily close powers of $a$ and $b$?

This post was motivated by this other post. I'm aware of Pillai's conjecture (yet to be proven) which states that the gaps in the sequence of perfect powers tend to infinity. However, what happens if ...
Alma Arjuna's user avatar
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1 vote
0 answers
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Product quotient of $SL_2(\mathbb{R})$ by non-commensurable discrete subgroups

This question is related to Kronecker's approximation theorem: fix a positive integer $N$ and consider $N$ real numbers $x_1,\cdots,x_N$ which are linear independent over $\mathbb{Q}$. Then the ...
Zhan's user avatar
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Existence of open interval $(x − δ_k, x + δ_k)$ which does not contain any rational number with denominator $k.$ [duplicate]

Let $x$ be an irrational number. Show that for any positive integer $k ≥ 1$, there exists $δ_k > 0$, such that the open interval $(x − δ_k, x + δ_k)$ does not contain any rational number of the ...
Abcd's user avatar
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6 votes
1 answer
178 views

Given two unbounded subsets $X,Y$ of $\mathbb{R},$ do there exist three points of $X$ whose translation and stretch approximates three points of $Y?$

Suppose $X$ and $Y$ each are subsets of $\mathbb{R}$ that are bounded below and unbounded above (and therefore infinite). Given $\varepsilon>0,\ $ do there exist $\ x_1,\ x_2,\ x_3 \in X;\ x_1 < ...
Adam Rubinson's user avatar
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Overview of "Some remarks on the Lonely Runner conjecture"

I'm reading this paper https://arxiv.org/pdf/1701.02048.pdf (Some remarks on the lonely runner conjecture - Terence Tao) and I think the maths is slightly out of my reach. Of chapter 3 I understand ...
crewmate's user avatar
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0 answers
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Rational approximation of $\sqrt[3]{2}$ for the diophantine $x^3 - 2y^3 = 1$ [duplicate]

In the book 'A Classical Approach to Modern Number Theory' by Ireland and Rosen, the penultimate paragraph of Chapter 17(Diophantine Equations) mentions, Thus while $x^2 - 2y^2 = 1$ has infinitely ...
Subham Jaiswal's user avatar
1 vote
0 answers
51 views

How fast does $\inf_{|a|, b|, |c|\leq n}|a+b\sqrt 2+c\sqrt 3|$ go to zero?

Given a real number $\alpha$ we can define its irrationality measure $\mu=\mu(\alpha)$ as the largest number such that $$\inf_{|a|, |b|\leq n} |a+b \alpha| \leq \frac{k}{n^{\mu+\epsilon}}$$ for some ...
Derivative's user avatar
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3 votes
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Calculating a lower bound of Lebesgue measure of some Borel set

I have the following set. I need to calculate a lower bound on its measure in order to prove something about some distribution. $I \subseteq [0,1)$. Take the set $\mathcal{J}(I) = I \bigcap \underset{\...
user2582354's user avatar
2 votes
0 answers
67 views

Finitely Many Solutions to Diophantine Equation

I am trying to prove the following claim: Let $F \in \mathbb{Z}[X, Y]$ be a square-free binary form of degree $d \geq 4$, and let $G \in \mathbb{Z}[X, Y]$ be a polynomial of total degree $g \leq d − 3$...
Bobo's user avatar
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2 votes
1 answer
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Proving an exponential function dominates over an erratic fractional part

Consider the function $3^x(1-2^{-\text{frac}(x\sqrt2)})$, where $\text{frac}()$ denotes the fractional part function. Now $\sqrt 2$ being irrational by kronecker, this is uniformly dense in $(0,1)$, ...
Us Prasad's user avatar
1 vote
0 answers
41 views

Example of the real numbers $x$ appearing in the Jarnik's theorem: $||nx||\leq n^{-\beta}$ for infinitely many $n\in\mathbb{N}$ for fixed $\beta>1$

According to Falconer (Falconer, '85, The Geometry of Fractal Sets, theorem 8.16, p. 134), a part of the Jarnik's theorem is the following: Take $\beta>1$. The set of real numbers $x$ for which ...
Epsilon Away's user avatar
1 vote
1 answer
68 views

Generalisation of Kronecker's Theorem

I am reading this [Six Lonely Runners]https://www.researchgate.net/publication/220343204_Six_Lonely_Runners, specifically Chapter 4. I do not understand their use of Kronecker's Theorem here and the ...
crewmate's user avatar
1 vote
2 answers
89 views

Liouville numbers and numbers having infinite irrationality measure

Let's start with the definition of a Liouville number: A Liouville number is a real number $\xi$ such that for any $m\in\mathbb N_>0$ there exists a pair of coprime integers $(p,q)\in\mathbb Z^2$ ...
manifold's user avatar
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3 votes
1 answer
364 views

Help understanding steps to prove Dirichlet's approximation theorem

In my elementary number theory textbook, there is a problem that is meant to help understand the proof of Dirichlet's rational approximation theorem. There are two parts to the problem: the first part ...
codeing_monkey's user avatar
2 votes
1 answer
75 views

Let $x\in\mathbb{R}.$ For each $n,$ define $x_n:=\min_{k\in\mathbb{Z}}\lvert\frac{k}{n}-x \rvert.$ For which values of $x$ does $\sum x_n$ converge?

Let $\ x\in\mathbb{R}.\ $ For each $\ n\in\mathbb{N},\ $ define $\ x_n := \displaystyle\min_{k\in\mathbb{Z}}\left\lvert \frac{k}{n} - x \right\rvert.\ $ For which values of $\ x\ $ does $\ \...
Adam Rubinson's user avatar
0 votes
1 answer
32 views

Three or more points moving on a circle and a convergent subsequence

Three (or more difficult version: any number $n>2$) points move on a unit circle with pairwise different constant speeds. For which initial points and which speeds is the following true: for every $...
Jkbb's user avatar
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0 answers
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Orbits Of $SL_d(\mathbb{Z})$ on non rational and non - irrational points in the $d$ - Dimensional Torus

On the $d$ - dimensional torus, for every irrational point $\theta = (\theta_1,...,\theta_d)$ (i.e $\theta_i \notin \mathbb{Q}$), the orbits of $SL_d(\mathbb{Z})$ on this point are dense. However, ...
user2582354's user avatar
3 votes
1 answer
127 views

Equality case in Dirichlet's approximation theorem

In its strong form, Dirichlet's approximation theorem (for dimension one) states that for $\alpha\in{\mathbb R}$, and an integer $n\geq 1$, we have $$ \min_{1\leq k\leq n}\| k \alpha \| \leq \frac{1}{...
Ewan Delanoy's user avatar
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5 votes
2 answers
198 views

How is Siegel's lemma applied in number theory?

In the Wikipedia page, Siegel's lemma is stated as follows: Consider the system $$ \begin{cases} \sum_{i=1}^Na_{1i}X_i=0\\ \vdots\\ \sum_{i=1}^Na_{Mi}X_i=0 \end{cases}, $$ where the coefficients $a_{...
mathslover's user avatar
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0 votes
0 answers
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Does a limit exist for the fraction of $p\alpha_i$?

With $p$ an integer and $\alpha_i$ a real number, does $\{p\alpha_i\}\leq\epsilon$ hold for some predefined $\epsilon$, maybe depending on the number of $\alpha_i$ values? I looked at the simultaneous ...
Jeroen Boschma's user avatar
0 votes
1 answer
164 views

Can we scale the elements of a set of real numbers so that they are arbitrarily close to integers?

Let $S$ be a finite set of real numbers. For any $\varepsilon > 0,$ is it always possible to choose $\alpha > 0$ such that every element of $S$ multiplied by $\alpha$ is withing $\varepsilon$ of ...
mathlander's user avatar
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1 vote
1 answer
159 views

Do three arithmetic sequences of real numbers that include $0$ always get arbitrarily close to each other at nonzero numbers?

I know that two arithmetic sequences of real numbers that both contain the number $0$ have terms that are arbitrarily close together, that is, for any $\varepsilon>0$ there exists a real number ...
mathlander's user avatar
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3 votes
0 answers
63 views

Absolute convergence of $\sum_{n=1}^\infty n^{-5} \csc(2n\pi \sqrt[3]{3})$

I want to prove the absolute convergence of this serie. There are two hints: Firstly, prove that $|(\sin(\pi x)| \geq |x|$ for $|x| \leq \frac{1}{2}$. Then use the fact that $2\sqrt[3]{3}$ is ...
AnonymousTNT's user avatar
2 votes
0 answers
48 views

Rational approximations of logarithms of ratios of primes

I noticed that there are many pairs of primes $p_1, p_2$ such that $$\frac {\log(p_1/p_2)}{ \log(2)} \approx \frac{N}{69324}$$ for some natural N with an error of order $10^{-8}$ . I found around 20 ...
Praveen B.'s user avatar
9 votes
2 answers
326 views

Are there any perfect squares of the form $\underbrace{88\cdots8}_{n\text{ times}}1$ (in decimal, at least two $8$'s)?

I saw this problem recently and it is deceptively hard. The usual $\pmod{4}$ trick won't work, and indeed there will be perfect squares whose last $n$ digits will be $\underbrace{88\cdots8}_{n\text{ ...
Alex Eustis's user avatar

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