For questions about approximating real numbers by rational numbers.

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12
votes
3answers
542 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 ...
25
votes
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.
1
vote
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
votes
2answers
186 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
votes
3answers
779 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
votes
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
569 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
vote
1answer
403 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
votes
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 ...