For questions about approximating real numbers by rational numbers.

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2answers
142 views

Constructive proof need to know the solutions of the equations

Observe the following equations: $2x^2 + 1 = 3^n$ has two solutions $(1, 1) ~\text{and}~ (2, 2)$ $x^2 + 1 = 2 \cdot 5^n$ has two solutions $(3, 1) ~\text{and}~ (7, 2)$ $7x^2 + 11= 2 \cdot 3^n$ has ...
3
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2answers
208 views

Solutions of $x^2 + 119 = 15 \cdot 2^n$ without trial and error

I seen this equation at math.stack exchange The equation $x^2 + 119 = 15 \cdot 2^n$ has only six solutions. Those are $(1,3) ,(11, 4), (19, 5), (29, 6), (61, 8)$ and other one is I don't know. This ...
8
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1answer
101 views

Do best lower approximations of a quadratic irrational always form a linear recurrence sequence?

Let $\theta$ be an irrational number and let $$ {\cal L}= \bigg\lbrace (a,b) \in {\mathbb Z} \times {\mathbb N}^{*} \bigg| \frac{a}{b} \leq \theta \bigg\rbrace $$ and $$ {\cal B}= \bigg\lbrace ...
13
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1answer
554 views

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 ...
4
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0answers
150 views

Is an integer in the interval $\left[\small {3^n+1 \over 2^n+1}\lt {3^n \over 2^n} \lt{3^n-1 \over 2^n-1}\right] $ for some $\small n>1$?

Consider the expression $$ f_n(j)= {3^n+j \over 2^n+j} $$ where we select some fixed $n \gt 1$ and let j vary over the reals from +1 to -1 . I'm concerned with the problem, whether in the interval ...
7
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1answer
148 views

badly approximated numbers on the real line form a meagre set

Let $S$ be the set of real numbers $x$ such that there exist infinitely many (reduced) rational numbers $p/q$ such that $$\left\vert x-\frac{p}{q}\right\vert <\frac{1}{q^8}.$$ I would like to prove ...
23
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2answers
857 views

Is there any real number except 1 which is equal to its own irrationality measure?

Is there any real number except $1$ which is equal to its own irrationality measure? If so, then what is the cardinality of the set of all such numbers? Is the set dense on any interval? Is it ...
12
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3answers
558 views

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 ...
26
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3answers
3k views

Proving that $m+n\sqrt{2}$ is dense in R

I am having trouble proving the statement: Let $S = \{m + n\sqrt 2 : m, n \in\mathbb Z\}$. Prove for every $\epsilon > 0$, The intersection of $S$ and $(0, \epsilon)$ is nonempty.
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1answer
132 views

For what $\lambda$ does the sequence $\left\{ \frac{k}{2\lambda} \right\}$ converge?

I am reading some materials on hypergeometric functions and their relation to general L-functions and the authors make use of the following "well-known elementary result from diophantine ...
2
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2answers
191 views

An experiment was performed n times. The exact success rate rounds to 85.8%. What is the minimum value of n?

An experiment was performed n times. The success rate was given as 85.8% Clearly, if there were 858 successes out of 1000 trials it would give that percentage. However, the percentages are rounded to ...
12
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3answers
784 views

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

I am having trouble calculating the following limit: $$\lim_{n \to \infty} \sqrt[n]{|\sin n|}\ .$$
10
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1answer
182 views

What is the sum of the squares of the differences of consecutive element of a Farey Sequence

A Farey sequence of order $n$ is a list of the rational numbers between 0 and 1 inclusive whose denominator is less than or equal to $n$. For example $F_6= ...
2
votes
2answers
573 views

Is $\sin^2 n$ bounded away from zero?

Can one find a number $m$ such that $\sin^2 n \geq m > 0 $ for all integers $n$? By continuity of $\sin x$, it is enough to say that $|n - k \pi| \geq m^{\prime} > 0$ for all integers $n,k$. ...
1
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1answer
422 views

Non trivial upper bound for an exponential sum

Suppose $h \in \mathbb{N}$, is there a known non trivial upper bound for $$\left| \frac{1}{n} \sum_{m=1}^n e^{2 \pi i h (2 \pi m)} \right|?$$
5
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1answer
178 views

Estimating $\#\{\{\alpha k\} < 1/\sqrt{k} : k \leq n\}$ for irrational $\alpha$

Suppose $\{\alpha n\}$ is the fractional part of $\alpha n$. Put $$A_{\alpha}(n) = \#\{\{\alpha k\} < 1/\sqrt{k} : k \leq n\}.$$ If $\alpha$ is irrational, can I find some constant $K$ such that ...