Questions tagged [diophantine-approximation]

For questions about approximating real numbers by rational numbers.

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3 votes
0 answers
27 views

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{\...
1 vote
0 answers
28 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 $g \leq d − 3$...
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2 votes
1 answer
31 views

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 $(0,1)$, ...
1 vote
0 answers
23 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 ...
1 vote
1 answer
39 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 ...
1 vote
2 answers
43 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 $(p,q)\in\mathbb Z^2$ ...
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3 votes
1 answer
141 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 ...
2 votes
1 answer
64 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 $\ \...
0 votes
1 answer
30 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 $...
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1 vote
0 answers
22 views

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, ...
0 votes
0 answers
12 views

Find the Lagrange number of a given infinite simple continued fraction.

This question arises in Ch. 2 of Martin Aigner's beautiful book Markov's Theorem and 100 years of the uniqueness conjecture (page 36, Remark 2.4). These definitions are from Aigner's book summarized ...
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3 votes
1 answer
107 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}{...
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5 votes
2 answers
99 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_{...
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0 votes
0 answers
21 views

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 ...
0 votes
1 answer
107 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 ...
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1 vote
1 answer
140 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 ...
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3 votes
0 answers
61 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 ...
0 votes
0 answers
35 views

How does Roth's Theorem show a number is transcendental

[Roth's theorem][1] says that for irrational algebraic number $\alpha$ and $\epsilon>0$, there are finitely many solutions $\frac{p}{q}$to this: $$\displaystyle \left|\alpha-\frac pq\right|<\...
0 votes
0 answers
25 views

The fractional part of a geometric sequence.

Prove or deny: for any given positive integer $ N $, there is a positive integer $ m\geq n $ such that $ \left\{ \frac{2^m}{\alpha} \right\} >1-\frac{1}{\alpha} $, where $\left\{x\right\}$ ...
2 votes
0 answers
42 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 ...
7 votes
1 answer
158 views

Are there any perfect squares of the form 88...81 (in decimal, at least two 8's)?

I saw this problem recently and it is deceptively hard. The usual mod 4 trick won't work, and indeed there will be perfect squares whose last n digits will be 88...81, for any n. I can show that if ...
1 vote
1 answer
55 views

Reversing the coefficients of the continued fraction expansion of a rational number

On this Wikipedia page about the Markov constant, there's a surprising theorem. Basically if $\alpha$ is a real number with continued fraction expansion $[a_0;a_1,a_2,...]$, the claim is that the ...
2 votes
0 answers
59 views

Measuring how good the best possible rational approximation of a real number is

The Markov constant of a real number $r$ is $$ \limsup_{d \to \infty} \frac{1}{|r-n/d|d^2} $$ where we choose the best possible $n$ for each corresponding $d$, e.g. $n = \text{round}(r\cdot d)$. This ...
4 votes
0 answers
53 views

Almost Diophantine approximation

We have an algebraic number $a$ and a real number $b$. Can the following inequality have infinitely many solutions for $n \in \mathbb{N}$? $$ \{an\} \in [b - \frac{1}{2^n}, b + \frac{1}{2^n}] $$ Where ...
3 votes
1 answer
64 views

Diophantine approximation theory and "logarithmic" rather than "linear" error

There is an enormous amount of literature on Diophantine approximations, including the general theory of continued fractions, the Stern-Brocot tree, the notion of "badly approximable number" ...
1 vote
0 answers
50 views

Given $\alpha\in \mathbb{R},$ do there exist arbitrarily large $m,k$ such that $\vert \alpha - \frac{k}{m}\vert < \frac{1}{m^2}$? [duplicate]

Given $\alpha\in \mathbb{R},\ n\in\mathbb{N}$ does there exist $m\in\mathbb{N}$ with $m>n,\ $ and $\ k\in\mathbb{Z}, $ such that $$ \alpha \in\left[\frac{k}{m}-\frac{1}{m^2}, \frac{k}{m} + \frac{1}{...
3 votes
0 answers
64 views

How well approximated by rationals are almost all real numbers?

Let $\psi:\Bbb{N}\to\Bbb{R}_{\geq 0}$ be such that $q\mapsto q\psi(q)$ is a (weakly) decreasing function. We say a real number $x$ is $\psi$-approximable if $|qx-p|<\psi(q)$ has solutions for ...
0 votes
0 answers
19 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
61 views

Growth rate of $|a^n+b^n|$ where $a,b=(1\pm\sqrt{-7})/2$.

For the real sequence $$V_n = |a^n+b^n| \quad\text{ with }\quad a=\frac12(1+\sqrt{-7}),\ \ b=\frac12(1-\sqrt{-7}) \tag1$$ I'd like to derive some estimates on the growth rate of $V_n$ and $V_{2^n}$. ...
0 votes
1 answer
80 views

Reduction step in the proof of Roth's theorem

I am following Hindry-Silverman's book "Diophantine Geometry - An Introduction". In the proof of Roth's theorem, there is a reduction step showing that the following two theorems are ...
2 votes
0 answers
52 views

Reference request: A. Schinzel on digital sums of powers

I am searching for the earliest published proof of the following result: $$\lim_{k\to\infty} s(2^k) = \infty$$ where $s(n)$ denotes the sum of the decimal digits of $n$. This problem has been ...
2 votes
1 answer
76 views

Least upper bound on $x_n = \min{|a - b\sqrt{3}| : a + b = n}$.

For all integers $n \geq 1$, define $x_n = \min{|a - b\sqrt{3}| : a + b = n}$ where $a$ and $b$ are positive integers. Find the smallest positive real number p such that $x_n \leq p$ for all $n \geq 1$...
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3 votes
1 answer
47 views

Why does $|x-p/q|\ge\frac{C(x)}{q^a}$ for all $p/q$ imply: "$|x-p/q|\lt\frac{1}{q^a}$ has only finitely many solutions"?

This might be quite a trivial question but this has been bugging me, I don't know what the resolution is. Context: the irrationality measure $\mu=\mu(x)$ of some real number $x$ is defined to be that ...
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1 vote
0 answers
32 views

Showing that linear fractional equivalences preserve the irrationality measure

Throughout this post, let $(a,b,c,d)$ refer to the entries of some unspecified element: $$\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\mathrm{GL}_2(\Bbb Z)$$By a “linear fractional equivalence” (...
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3 votes
1 answer
56 views

Approximating a number by using only a given set of prime-factors for the approximant.

Let $n \in\mathbb{N},$ and $S$ be a (finite) set of prime numbers. I'm looking for an efficient algorithm to find the greatest $m\leq n$ such that $m$'s prime factors are of $S$? For $S=\{p\}$ the ...
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3 votes
1 answer
104 views

Simple estimation of separation of powers of 2 and powers of 3?

An accepted answer is in the cross post at https://mathoverflow.net/questions/428396/simple-estimation-of-difference-of-powers-of-2-and-powers-of-3 . 1. Question How to get from the formulas $$ \left| ...
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1 vote
1 answer
60 views

Rational distances to 3 points in ${\bf R}^2$

Consider three points $(0,0)$, $(1,0)$, and $(0, {\alpha})$ in ${\bf R}^2$. If there exists a point $P =(x,y)$ such that $x^2 + y^2 =r^2$, $(x-1)^2 + y^2 =s^2$, and $x^2 + (y-{\alpha})^2 =t^2$ where $...
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2 votes
1 answer
106 views

Approximating reals by p-smooth ratios

Is there a techinque that allows us to approximate, with arbitrary precision, any positive real number $x$ by a sequence of ratios of $p$-smooth numbers, for given prime number $p>2$? If yes, is ...
0 votes
0 answers
30 views

$|e^{inr}-1| \geq C>0$, $\forall n \in \mathbb{Z}$

I'm looking for an optimal assumption on $r$ such that $|e^{2inr}-1| \geq C>0$, for any $n$. It is clear that $$|e^{2inr}-1|^2=2(1-\cos(2nr)).$$ If $r$ is irrational, $1-\cos(2nr)>0$ but it is ...
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0 votes
1 answer
50 views

How fast can a series of positive rationals converge if we know its limit is rational?

Question: Let $\{x_n\}_{n\ge 0}$ be a sequence of positive reals for which $\sum_{n\ge 0}x_n$ converges. Does there always exist a sequence $\{y_n\}_{n\ge 0}$ of rationals such that $0<y_n<x_n$ ...
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5 votes
1 answer
135 views

How does Wolfram Alpha find polynomial equation of given roots?

I am experimenting with a method which will converge hopefully to a real number, for which I suspect, that it is the root of a polynomial equation. How does Wolfram Alpha find its guess? How does WA ...
1 vote
1 answer
49 views

Can $c^2(a\cdot b)+c(a+b)=2^c-2$ be solved to find $\mathbb{N}$ solutions?

$c^2(a\cdot b)+c(a+b)=2^c-2$ is of the form $mx+ny=k$ and should open the door to Diophantine, but do the constraints $x=(a\cdot b), y=(a+b)$ make a difference when trying to solve for $\mathbb{N}$ ...
-1 votes
2 answers
101 views

How can I find natural number solutions to $\frac{186-x}{11x+1}=y$? [duplicate]

I'm trying to find solutions to $\frac{186-x}{11x+1}=y$, where $x,y \in \mathbb{N}$. I've been researching Diophantine equations to try and solve this, but everything I've found is in the format $ax + ...
5 votes
1 answer
77 views

Is $(1+c^2)^n-\lfloor(1+c^2)^{n/2}\rfloor^2<(1+c^2)^{(n+1)/2}$ true for all integers $c>1$, when $n$ is an odd integer?

Let $n$ be an odd integer. Is $$(1+c^2)^n-\lfloor(1+c^2)^{n/2}\rfloor^2<(1+c^2)^{(n+1)/2}$$ true for all integers $c>1$? Notes: $c=1$ has a counterexample $2^{31}-\lfloor2^{31/2}\rfloor^2>2^{...
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3 votes
3 answers
89 views

Does $\lfloor \tan (n)\rfloor =n$ have infinite roots?

Background This post can be understood as the gap between $\tan n$ and $n$ has no upper limit. Does the sequence $n+\tan(n), n \in\mathbb{N}$ have a lower bound? I want to know how close $\tan n$ can ...
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1 vote
1 answer
118 views

If $\Big|x-\frac{p}{q}\Big|<\frac{1}{2q^2}$ then $p/q$ is necessarily one of the convergents : Extend the proof to irrational $x$

Prove that, if $x$ is any irrational number, and if $p/q$ is a rational fraction in lowest terms, with $q\geq 1$, such that $$\Big|x-\frac{p}{q}\Big|<\frac{1}{2q^2}$$ then $p/q$ is necessarily one ...
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1 vote
0 answers
26 views

Validity of Equality in the Condition for a Rational to be a Convergent

Let $x$ be an irrational number and the rational number $𝑎/𝑏$ satisfy the inequality: $$ \bigg|𝑥−\frac{a}{b}\bigg|<\frac{1}{2b^2} $$ Then $𝑎/𝑏$ is a convergent of $x$ A proof by contradiction ...
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0 votes
1 answer
69 views

Rational solutions of the equation $2xy^2=kx^2+1$

I am looking for the rational solutions of the equation: $$2xy^2=kx^2+1$$ where $k$ is a fixed positional natural number. For $k=6$, Maple shows it is irreducible and doesn't produces the rational ...
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1 vote
1 answer
50 views

Does every line through the origin come arbitrarily close to some other lattice point?

Given positive real numbers $x$ and $\epsilon$, do there necessarily exist positive integers $p$ and $q$ such that $\left| qx - p \right| < \epsilon$? Or geometrically: in a two-dimensional ...
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2 votes
0 answers
37 views

Range of Lagrange's 'best approximations' law

In my reference the theorem is stated as, convergents to an (irrational) number give a sequence of best approximations. It is also given a proof by contradiction : Assuming $|q\alpha-p|<|q_n\alpha-...
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