Questions tagged [diophantine-approximation]

For questions about approximating real numbers by rational numbers.

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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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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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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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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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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
30 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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  • 607
2 votes
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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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Lagrange's best approximations law proof by contradiction doubt

In the proof of Lagrange's best approximations law it is stated that the system of equations $$ xp_n+yp_{n+1}=p\\ xq_n+yq_{n+1}=q\\ $$ where $\dfrac{p_n}{q_n}$ is the rational is a convergent of the ...
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lonely runner conjecture for 3 runners

I am looking at the lonely runner conjecture and it seems that people often prove that the stationary runner is lonely for 3 runners. Why is it not considered when the other runners are lonely?
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Why is $|dx-c|<|qx-p|$ the condition for the best rational approximations of the second kind

The fraction $c/d$ is a best approximation of the second kind for a number $x$ if for every other fraction $p/q$ with $q<d$, $|dx-c|<|qx-p|$ Given a number $x$, its best rational approximations ...
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  • 6,801
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Is the convergence of these two series equivalent? (They come from Khinchin's theorem and the Duffin-Schaeffer conjecture.)

I am trying to wrap my head around two theorems of Diophantine approximation: Khinchin's theorem and the Duffin and Schaeffer conjecture. To the best of my understanding, here is what they say: ...
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1 vote
1 answer
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Difficulty with diophantine approximation: building sets with arbitrary lower and upper natural densities

$\newcommand{\d}{\mathrm{d}}\newcommand{\du}{\overline{\mathrm{d}}}\newcommand{\dl}{\underline{\mathrm{d}}}\newcommand{\card}{\operatorname{card}}$This was left as an exercise: Show that for all $0\...
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83 views

Upper and lower bounds for $n|\sin(n)|$

In asymptotic analysis, we have the following set of functions \begin{align} O(g) &= \big\{ f:\mathbb{N}\to\mathbb{R} \,|\, \exists C \in \mathbb{R}^+, \, \exists N \in \mathbb{N}^+, \, \forall n\...
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4 votes
2 answers
77 views

Relationship between Dirichlet's Approximation Theorem and Convergents

For any real number $r$, the convergents to the continued fraction expansion of $r$ satisfy Dirichlet's approximation inequality of $|r - \frac{p}{q}| < \frac{1}{q^2}$. Does this go the other way? ...
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2 votes
2 answers
144 views

Is there a number such that the sequence of its best rational approximations is strictly increasing?

The rational approximations of $\sqrt 2$ given by its continuous fraction are: 1.5, 1.333, 1.4, 1.417, 1.412, etc. which is not strictly increasing. Similarly, this sequence for $\phi=(1+\sqrt5)\big/2$...
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Generalizing how well an arbitrary set approximates a number?

Suppose $A\subseteq\mathbb{R}$ and $x\in \mathbb{R}$. I want a measure $\mu(x,A)$ that determines "how well" $x$ can be approximated by $A$? For example, if $A=\mathbb{Q}$, we use the ...
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1 vote
3 answers
106 views

For a perfect square $n$, can we calculate integers $0<y<l$ up to a limit $l$ such that $n+y^2$ is a perfect square (without bruteforce)?

I want to implement a fast algorithm (avoiding or at least minimizing bruteforce) which for a given square number $n$ calculates a series of positive integers $y$ up to a certain limit $l$ such that $...
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1 answer
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For a given integer $n>0$, can we efficiently determine whether an integer $y$ exist such that $n+y^2$ is a perfect square?

I am trying to implement an efficient algorithm which must determine whether for a given integer $n>0$ another integer $y$ exist such that $n+y^2$ is a perfect square. Based on some properties of $...
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4 votes
0 answers
94 views

Why are there finitely many of these approximations of $\sqrt{2}$? [duplicate]

I'm solving the following problem from a discrete geometry book (Lectures on Discrete Geometry, Jiri Matousek). Prove that for $\alpha=\sqrt{2}$ there are only finitely many pairs $m,n\in\mathbb{N}$ ...
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2 answers
160 views

Notice that $\sqrt{51}\approx 7+\frac{\sqrt{2}}{10}$

I was doing some other stuff and noticed that: $$\sqrt{51} - \left(7 + \dfrac{\sqrt{2}}{10}\right) = 0.0000070723\,\, (*)$$ and this immediately made me think of their respective continued fractions, ...
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2 votes
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Approximating irrational numbers with numbers in $\mathbb{Q}(\sqrt{5})$

I am trying to evaluate an infinite series that involves Fibonacci numbers. By adding the first thousands of terms, I get an approximation of the sum: $S = 5....
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  • 395
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1 answer
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Convergence of Fourier series: which kind of convergence is being used here?

I'm trying to understand the proof of Lemma 9 from Sprindzhuk's book Metric Theory of Diophantine approximation. Here's what it says. Lemma 9. Fix integers $m$ and $n$ and consider the tori $\mathbb{T}...
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1 vote
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32 views

One-sided Dirichlet approximation

A version of Dirichlet's Theorem states: Given $d \in \mathbb{R}$ and an integer $N \geq 2$, there exist integers $p$ and $q$ with $p > 0$ such that $$ |pd - q | \leq \frac{1}{N}.$$ Is the result ...
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0 answers
22 views

Diophantine condition for linearization of power series

I have been reading Siegel's "Iteration of analytic functions" (https://www.jstor.org/stable/1968952?seq=1#metadata_info_tab_contents), in the paper, it is stated that for a power series $f(...
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1 answer
86 views

Hurwitz's theorem for Diophantine approximation

Hurwitz's Theorem in Number Theory states that for every irrational number $\xi$, there are infinitely many relatively prime natural numbers $(p,q)$ satisfying the equation: $$ | \xi−\frac{p}{q}| < ...
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Why is irrationality mesure of irrational numbers at least 2

For any $x \in \mathbb{R}$, define its irrationality measure $\mu(x)$ to be the smallest number $\mu$ such that the inequality $|x-\dfrac{p}{q}| > \dfrac{1}{q^{\mu+\epsilon}}$ holds for any $\...
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1 vote
1 answer
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How to approximate a rational number $m/n$ with fractions $a/b$ such that the product $ab$ is given

Consider a certain rational number $\alpha=m/n$ and let $N$ be a positive integer. Is there a way to find the better rational approximation $x/y$ of $\alpha$ between all the fraction such that $xy=N,$ ...
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3 votes
0 answers
81 views

How many simple continued fractions have $p_k=n$?

Let $\alpha$ be a number with continued fraction $[a_0; a_1, a_2, ...]$, and let $\frac{p_k}{q_k}=[a_0;a_1,a_2,...,a_k]$ be the $k$-th convergent. I'm interested in how many different fractions ...
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2 votes
1 answer
60 views

Approximation of a real number via a fraction of coprimes.

I'm reading a paper on number theory (which is not my field at all) stating, without any proof, a claim which can be rephrased as Fix a positive integer $M$. Then, given any real number $\alpha$, ...
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Diophantine Approximations of high dimensions

I am struggling with this question [enter image description here][1] [1]: https://i.stack.imgur.com/EIvo5.pngstrong text Where Vol_m or Vol_n is the lesbegue measure of the unit ball I would realy ...
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2 votes
0 answers
61 views

Ratios of integers in Bezout's Identity

Bezout's Identity is a classic of elementary number theory: let $m,n\in\mathbb{N}^+$ with $\gcd(m,n)=1$. Then there are $a,b\in\mathbb{Z}$ with $$ am+bn=1 $$With no loss of generality we can assume $m&...
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1 vote
1 answer
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Understanding Minkowski's estimate for the product of successive minima from Cassel's book

I wonder if there is a mistake in the proof of Theorem IV in section VIII.4.2 in Cassel's book on geometry of numbers. I refer to the claim in equation (12) on page 217 and its proof three equations ...
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0 answers
51 views

Estimating the number of solutions for a quadratic inequality.

It is well known in the literature that if $0< A, E < N^{O(1)}$ ($A, E$ integers) and $A$ is not a perfect square, then the inequality $$ \left|x^2 + A y^2 - E\right| \leq \frac{1}{2} \qquad (\...
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4 votes
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127 views

Finding a pair $(a,b)\in \mathbb R^2$ such that $\inf_{m,n\in \mathbb Z}\left|(m+\sqrt{3}n+a)(\sqrt{2}m+n+b) \right|>0$

I am trying the find a pair $(a,b)\in \mathbb R^2$ such that $\inf_{m,n\in \mathbb Z}\left|(m+\sqrt{3}n+a)(\sqrt{2}m+n+b) \right|>0$ For $a=b=\frac{1}{2}$, I speculate that the infimum is nonzero. ...
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5 votes
1 answer
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The lower bound of $|3^p - 2^q|$ - how to derive from Baker's theorem?

In his blog, Terence Tao discussed the lower bound of $\vert 3^p - 2^q \vert$ in the following corollary. Corollary 4 (Separation between powers of $2$ and powers of $3$) For any positive integers $p$,...
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1 vote
1 answer
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Best rational approximations to a ratio of three numbers $x:y:z$

If I have a ratio of two positive real numbers $x:y,$ then I can find the "best" rational approximations to it by writing it as a continued fraction (e.g. by repeatedly removing the integer ...
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  • 111
1 vote
2 answers
134 views

Finding the square root of a polynomial with radical

I have two positive integers, $a$ and $b$, such that $$a^2=28b\sqrt{8b^2+1}+80b^2+5, \tag{$\star$}$$ and I’d like to find $a$ in terms of $b$ (including $\sqrt{8b^2+1}$, if necessary/appropriate), ...
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24 votes
5 answers
1k views

Does there exist $\ n,m\in\mathbb{N}\ $ such that $\ \lvert \left(\frac{3}{2}\right)^n - 2^m \rvert < \frac{1}{4}\ $?

Does there exist $\ n,m\in\mathbb{N}\ $ such that $\ \lvert \left(\frac{3}{2}\right)^n - 2^m \rvert < \frac{1}{4}\ $ ? I have tried for the first few integers $\ n,m\ $ up until $\ m\approx30\ $ ...
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5 votes
0 answers
131 views

A Gap in the Proof of the Duffin-Schaeffer's theorem

Let $\varphi$ be the Euler totient function and $f$ be a nonnegative function such that $$\sum_{n=1}^{\infty} \frac{f(n)\varphi(n)}{n}=\infty,$$ and that $$\limsup_{N\to \infty} \frac{\sum_{n=1}^N f(n)...
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1 vote
0 answers
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Measure or irrationality and discriminant

For an algebraic number $\alpha$ of degree $2$, I know how to prove that there is a constant $c>0$ such that $$\left|\alpha - \frac{p}{q}\right| \geq \frac{c}{q^2}.$$ However, I get the constant $c$...
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1 vote
0 answers
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How to approxmate a real number by a sum of two real with integer weights?

Given three real numbers, $a$, $b$, and $c$, I am wondering how to approximate $c$ by the sum $an+bm$ for integers $n$ and $m$ up to the desired accuracy. More precisely, define $z:\mathbb{Z}\times \...
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3 votes
1 answer
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What did Thue prove when, concerning rational approximations?

A follow-up to When did Liouville come up with the first transcendental numbers? In 1909, Thue showed that if $\alpha\in\mathbb R$ is algebraic of degree $n$ and $s>\frac12n+1$, and if $c$ is any ...
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  • 77k
2 votes
1 answer
63 views

Pseudo-lonely runner conjecture with $\frac{1}{k+1}$ and generalizations

I was reading about the unsolved lonely runner conjecture on Wikipedia, which states "[c]onsider $k$ runners on a circular track of unit length. At $t=0$, all runners are at the same position and ...
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4 votes
3 answers
157 views

Does $\min |{\cos(n)}|$ exist?

Let $f:\mathbb{Z^+}\rightarrow \mathbb{R}$ where $f(n)=|\cos(n)|$ in radians. Does $\min f$ exist? I think the answer is no and I that have the right approach to proving it. From Dirichlet ...
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5 votes
0 answers
35 views

Chromatic polynomial of the cross-polytope and denominators of convergents to e.

Let $C_n$ denote the $1$-skeleton of the $n$-dimensional cross-polytope, and $\chi_{C_n}(x)$ be the chromatic polynomial of $C_n$. This is equivalent to the way of coloring the $(n-1)$-dimensional ...
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2 votes
1 answer
96 views

Finding an integer vector perpendicular to a rational vector/solving a Diophantine equation with constraints.

Essentially, I want to solve the following problem: Given a vector of rational numbers, $\vec{X}\in \mathbb{Q}^d$, find the smallest integer vector perpendicular to it. That is, find $\vec{n}\in \...
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  • 123
3 votes
0 answers
49 views

How are almost integers of the form $(n^2-1)\pi$ related to convergents?

Two fractions related to early convergents of $\pi$ are the semi-convergents $$\frac{333-22}{106-7}=\frac{311}{99}$$ and $$\frac{355+22}{113+7}=\frac{377}{120}$$ The denominators of these fractions ...
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10 votes
1 answer
294 views

355/113 and small odd cubes

An important approximation to $\pi$ is given by the convergent $\frac{355}{113}$. The numerator and the denominator of this fraction are at the same distance of small consecutive odd cubes. $$\frac{...
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2 votes
0 answers
28 views

Effective equidistribution for number with bounded irrationality measure

This question asked for bounds greater than $\omicron\left(n\right)$ on the error $$ E_n=|T\cap\{1,2,\ldots\}|-\ell n. $$ where $$ \ell=\lim_{n\to\infty}\frac{|T\cap\{1,2,\ldots\}|}{n} \qquad\text{ ...
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  • 176
4 votes
1 answer
63 views

Convergent subsequences of $\{ n \alpha \bmod 1 \}$

It's well known that if $\alpha \not\in \mathbb{Q}$, then the sequence $\{n \alpha \bmod 1\}_{n \geq 0}$ is dense in the torus $\mathbb{T} = \mathbb{R}/\mathbb{Z}$. Does every convergent subsequence ...
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