Questions tagged [diophantine-approximation]

For questions about approximating real numbers by rational numbers.

432 questions
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Can we make $\tan(x)$ arbitrarily close to an integer when $x\in \mathbb{Z}$?

My 7-year-old son was staring at the graph of tan() and its endlessly-repeating serpentine strokes on the number line between multiples of $\pi$ and he asked me the question in the title. More precisely, is the following true or false? For any…
Fixee
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How to find $\sum_{i=1}^n\left\lfloor i\sqrt{2}\right\rfloor$ A001951 A Beatty sequence: a(n) = floor(n*sqrt(2)).

A001951 A Beatty sequence: a(n) = floor(n*sqrt(2)). If $n = 5$ then $$\left\lfloor1\sqrt{2}\right\rfloor+ \left\lfloor2\sqrt{2}\right\rfloor + \left\lfloor3\sqrt{2}\right\rfloor +\left\lfloor4 \sqrt{2}\right\rfloor+…
18
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Does $ \sqrt[n]{\left\lvert \sin(2^n) \right\rvert} $ have a limit?

The entire question is essentially given in the title: For $ n $ a positive integer, does the sequence $ x_n = \sqrt[n]{\left\lvert \sin(2^n) \right\rvert} $ have a limit as $ n \to \infty $? Background: One of the questions in a recent batch of…
18
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Why is an irrational number's algebraic complexity the opposite of its Diophantine complexity?

Definition 1. Given $x \in \Bbb{R}$, the algebraic degree of $x$ is the degree of the minimal polynomial of $x$ over $\Bbb{Q}$. If $x$ is transcendental, we will define its algebraic degree to be $\infty$. Definition 2. Given $x \in \Bbb{R}$, the…
18
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2 answers

Exploiting a Diophantine approximation of $\pi^4$ into giving a series of rationals for $\pi^4$

A note about this question: The original question asked seems likely impossible so I am really asking if we can exploit the technique below into giving us a 'nice' form for $\pi^4$. By nice form I mean an explicitly defined series of rationals. A…
17
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1 answer

Does the sequence $\{\sin^n(n)\}$ converge?

Does the sequence $\{\sin^n(n)\}$ converge? Does the series $\sum\limits_{n=1}^\infty \sin^n(n)$ converge?
16
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5 answers

Drawing approximated regular shapes on square grid

I find myself often fooling around with pen and paper, preferably squared paper. So I began looking for ways to sketch geometric figures as precisely as possible without using compass and/or ruler. In particular I'm thinking of regular polygons and…
15
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3 answers

Calculate $\lim_{n \to \infty} \sqrt[n]{|\sin n|}$

I am having trouble calculating the following limit: $$\lim_{n \to \infty} \sqrt[n]{|\sin n|}\ .$$
15
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1 answer

Is there a 'far' irrational number?

I recently learned of so-called 'far' numbers at a talk. In the talk, it was proven that there is a dense subset of the interval $[0,1]$ of far numbers (however, far numbers were only a minor point of the talk, so we quickly went on to bigger and…
davidlowryduda
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When is a sequence $(x_n) \subset [0,1]$ dense in $[0,1]$?

Weyl's criterion states that a sequence $(x_n) \subset [0,1]$ is equidistributed if and only if $$\lim_{n \to \infty} \frac{1}{n}\sum_{j = 0}^{n-1}e^{2\pi i \ell x_j} = 0$$ for ever non-zero integer $\ell$. I was wondering if anyone knows of a…
14
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In Search Of Elementary Proof Of Kobayashi's Theorem

There is a theorem in Number Theory due to Hiroshi Kobayashi (possibly less famous). The statement of this theorem is quite simple-looking. The original proof of Kobayashi relies on Siegel's Theorem in Diophantine Geometry, which is a deep theorem…
14
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Can every algebraic integer of degree $3$ be approximated by a quotient of linearly recurrent integer sequences of degree $3$?

Given a zero $\alpha \in \mathbb{R}$ of an irreducible monic third degree polynomial $x^3 - a_2x^2 - a_1x - a_0$, are there always integer sequences $(p_n)_{n=1}^\infty$ and $(q_n)_{n=1}^\infty$ satisfying recursions $$p_{n+1} = b_2p_n + b_1p_{n-1}…
14
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3 answers

A series to prove $\frac{22}{7}-\pi>0$

After T. Piezas answered Is there a series to show $22\pi^4>2143\,$? a natural question is Is there a series that proves $\frac{22}{7}-\pi>0$? One such series may be found combining linearly the series that arise from truncating…
14
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Accumulation points of $ \{x_n \in \mathbb{R}, n \in \mathbb{N} \ \ | \ x_n = n\sin(n) \}$?

A younger student asked me: What are accumulation points of the following set? $$ \{x_n \in \mathbb{R}, n \in \mathbb{N} \ \ | \ x_n = n\sin(n) \}$$ I really can't answer this question, could anyone help me?
Ivan Di Liberti
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How was this approximation of $\pi$ involving $\sqrt{5}$ arrived at?

The Wikipedia article for Approximations of $\pi$ contains this little gem: $$ \pi \approx \frac{63}{25}\times\frac{17 + 15\sqrt{5}}{7 + 15\sqrt{5}} $$ which is clearly in $\mathbb{Q[\sqrt{5}]}$. Wikipedia doesn't (currently) give a reference for…
hatch22
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