198
votes

Accepted

### Is the series $\sum_n|\sin n|^n/n$ convergent?

Note that if $\pi$ were rational (with even numerator), then $\sin(n)$ would equal $1$ periodically, so the series would diverge. Similarly if $\pi$ were a sufficiently strong Liouville number. Thus,...

40
votes

### Is the series $\sum_n|\sin n|^n/n$ convergent?

On the OP request, here is the plot of first 10000 partial sums.
Following Terry Tao's suggestion, here is the plot of ($n$th partial sum) $+2^{\frac32}/\sqrt{\pi n}$ for $n$ up to one million:
The ...

Community wiki

28
votes

Accepted

### Simultaneous Diophantine approximation of $\sqrt{2}$ and $\sqrt{2\pm \sqrt{3}}$

$$\sqrt{2+\sqrt{3}}-\sqrt{2- \sqrt{3}}=\sqrt{2}$$

27
votes

### For $x$ irrational, is $a_{n} =\sum_{k=1}^{n}(-1)^{⌊kx⌋}$ unbounded?

It's indeed unbounded for every irrational $x$.
Let me identify points of $\mathbb{R}/\mathbb{Z}$ with their representatives on $[0,1)$, and order it by the usual order $<$ of $\mathbb{R}$ applied ...

26
votes

Accepted

### $\lfloor \frac{3^n}{2^n} \rfloor = \lfloor \frac{3^n-1}{2^n-1} \rfloor$ for $n\geq2$

Mahler proved in 1957 (see here) that if $q$ is a positive rational number which is not an integer, then the distance of $q^n$ to the nearest integer is $(1-o(1))^n$. In particular, taking $q=3/2$, we ...

24
votes

Accepted

### Does $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converge to zero when $\alpha$ is irrational?

The product does not tend to the limit zero. For any irrational number $\alpha$ one can show that
$$
\limsup_{N\to \infty} \prod_{n=1}^{N} |1- e(n\alpha)| = \infty. \tag{1}
$$
(Here I use the usual ...

20
votes

Accepted

### Are there any solutions to the diophantine equation $x^n-2y^n=1$ with $x>1$ and $n>2$?

Delone (1930) and Nagell (1928) showed for any nonzero integer $d$ that the equation $x^3 - dy^3 = 1$ has at most one solution in integers $(x,y)$ besides $(1,0)$, with no constraint on the signs of $...

20
votes

### Is the series $\sum_n|\sin n|^n/n$ convergent?

Semilog plot building on მამუკა ჯიბლაძე's picture, this time to $10^7$

Community wiki

19
votes

### Source and context of $\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 dx/(1+x^2)$?

This integral has a series counterpart
$$\sum_{k=0}^\infty \frac{240}{(4k+5)(4k+6)(4k+7)(4k+9)(4k+10)(4k+11)}=\frac{22}{7}-\pi$$
https://math.stackexchange.com/a/1657416/134791
(UPDATE Peter Bala ...

19
votes

Accepted

### Can two rational rotations $F_2 = \langle A, B \rangle \to SO(3)$ efficiently approximate the $3 \times 3$ identity matrix?

Setting $a_0=A^7, b_0=B^7$ and then $a_{n+1}=[b_n^{-1},a_n], b_{n+1}=[a_n,b_n]$ seems very efficient. The length grows like $C \alpha^n$ with $\alpha= \frac{3 + \sqrt{17}}2$ and the operator norm ...

16
votes

### Rational approximations of $\sqrt{2}$ in $\mathbb{R} \times \mathbb{Q}_7$

Squares of elements of the $3$-adic integers $\mathbb{Z}_3$ are congruent to $0$ or $1$ modulo $3$, thus they are all at $3$-adic distance $1$ from $2$.
Squares of elements of $\mathbb{Q}_3 \setminus ...

16
votes

### Digits in an algebraic irrational number

What is known is that every real irrational has a $0$ in its $g$-ary expansion for infinitely many $g$. WLOG take $0 < x < 1$.
Taking an even-numbered convergent of the continued fraction of $...

15
votes

### Does $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converge to zero when $\alpha$ is irrational?

I have no idea how to approach the general problem, but here is a quick observation:
A. Let $\alpha = 2\beta$ so that $\beta$ is irrational if and only if $\alpha$ is so. Define $f$ by $f(x) = \log|2\...

15
votes

Accepted

### Upper bound for Hall's conjecture on separation of squares and cubes

The best $\theta$ is $0$. It is known that there are infinitely many
solutions of 0 < $|x^3 - y^2| \ll x^{1/2}$, parametrized by certain
"Pell equations"; indeed one such family attains $...

14
votes

Accepted

### Baker's theorem for integer combinations of logarithms of integers?

The main difficulty in proving Baker's theorem is in estimating $C$. If you don't care about those estimates, then the proof is not difficult. For example, Chapter 7 of Waldschmidt's Diophantine ...

13
votes

### Is the series $\sum_n|\sin n|^n/n$ convergent?

Let $D_N$ be the discrepancy:
$$
D_N=\sup \left| \frac{ A(J:P)}{N} - |J|\right|
$$
where $P=\{k/\pi \ \mathrm{mod} \ 1\}_{k=1,2,\ldots, n}$, $J$ is an interval in $[0,1]$.
If the irrationality ...

13
votes

### For $x$ irrational, is $a_{n} =\sum_{k=1}^{n}(-1)^{⌊kx⌋}$ unbounded?

We have
Theorem. Let $\psi(x)$ and $\varphi(x)$ be positive increasing functions such that
$$\int_1^\infty \frac{dx}{\psi(x)}=+\infty,\qquad \int_1^\infty \frac{dx}{\varphi(x)}<+\infty.$$
Then ...

13
votes

Accepted

### Continuous variant of the Chinese remainder theorem

Your guess is correct. For $\alpha = (\alpha_1, \alpha_2, \ldots, \alpha_d) \in \mathbb{R}^d$, the values of $m \alpha \bmod 1$ are equidistributed (and in particular dense) in $(\mathbb{R}/\mathbb{Z})...

12
votes

### Estimate number of solutions in the Roth's theorem

For a fixed $\alpha$, the number $N_{\alpha}(\epsilon)$ is bounded by a polynomial function of $1/\epsilon$. The proof of this requires either Faltings's product theorem, or Esnault and Viehweg's ...

12
votes

Accepted

### Rational approximations of $\sqrt{2}$ in $\mathbb{R} \times \mathbb{Q}_7$

Here is a full solution for the modified problem, inspired by Gro-Tsen's valuable comment.
1. There are infinitely many rational numbers $a/b\in\mathbb{Q}$ in lowest terms such that
$$ \left|\frac{a^...

11
votes

### Estimate number of solutions in the Roth's theorem

I believe that there is a completely explicit upper bound for $N_\alpha(\epsilon)$ (more generally counting in relative number fields and using more then one, possibly non-archimedean, absolute value) ...

11
votes

### Approximating any integer by multiples of 2 and 3

Fix $\varepsilon>0$. We have to prove (for large enough $n$) that there exist non-negative integers $a,b$ such that $\log_2 n\leqslant a+b\log_2 3<\log_2 n+\varepsilon$. This follows from the ...

11
votes

### A naive diophantine approximation question

The answer is no.
Start with $\alpha$ between 3.25 and 3.75; take squares of those 2 to see that one can restrict $\alpha^2$ to being between 11.25 and 11.75; take 3/2 powers of those to see that one ...

11
votes

### Lower bound for the fractional part of $(4/3)^n$

A non-trivial lower bound can be obtained using linear forms in $p$-adic logarithms. Suppose that $\left\{\left(\frac{4}{3}\right)^n\right\}$ is small. Clearly it is a rational number with denominator ...

11
votes

Accepted

### number of integers $n$ with $\|n \alpha \|$ small?

Yes. Assume that $\alpha$ is irrational, and its continued fraction digits do not exceed $K$. Then, for any positive integer $q$, we have
$$q\|q\alpha\|>1/(K+2).$$
In particular, for $q\leq N$, we ...

10
votes

### Does $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converge to zero when $\alpha$ is irrational?

Write it as $a_n(\alpha)$ to emphasize the dependence on $\alpha$.
For any $\epsilon > 0$, $U(n,\epsilon) = \{\alpha: |a_n(\alpha)| < \epsilon\}$ is an open set containing $k/m$ for any integers ...

10
votes

Accepted

### Growth of a linear recurrent sequence

First note that $\frac{1+i\sqrt{2}}{1-i\sqrt{2}}\in\mathbb{Q}(\sqrt{-2})$ is not a root of unity, because it does not equal $\pm 1$. Therefore Baker's famous theorem shows that, for some effectively ...

10
votes

Accepted

### Sign in Dirichlet's approximation theorem

Yes, this follows from considering the continued fraction of $\alpha$. If $p_n/q_n$ is the $n$th convergent to $\alpha$ and $n$ is odd then
$$ 0\leq \alpha - \frac{p_n}{q_n} \leq \frac{1}{q_n^2}.$$

10
votes

### For $x$ irrational, is $a_{n} =\sum_{k=1}^{n}(-1)^{⌊kx⌋}$ unbounded?

Here is an argument essentially due to fedja I learned about thirteen years ago on artofproblemsolving.com.
Proposition: if $f$ is $2$-periodic Riemann integrable such that $\sup_{n \geq 1} \left|\...

10
votes

### Well known applications of Roth's theorem

If nothing else we can use Roth's theorem to generalize
Liouville's construction of transcendental numbers.
Liouville noted that numbers such as $\lambda = \sum_{k=1}^\infty 1 / 10^{k!}$
are ...

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